4  Applications of Derivatives

4.1 Definition of Directional Derivative

Partial derivatives find the rate of change of \(z=f(x,y)\) in the directions of the \(x\) and \(y\) axis; that is in the direction of the unit vectors \(\vec{i}\) and \(\vec{j}\), respectively.

Definition 4.1 Let \(f\) be a function of two variables, and let \(\vec{u}=u_1\vec{i}+u_2\vec{j}\) be a unit vector. The directional derivative of \(f\) at \(P(a,b)\) in the direction of \(\vec{u}\) is defined by \[ D_{\vec{u}} f(a,b)=\lim _{h\to 0}\frac{f\left(a+h u_1, b+h u_2\right)-f(a,b)}{h} \] provided the limit exists.

::: {#thm- } [Directional Deriviative]

Let \(f(x,y)\) be a function that is differentiable at \(P(a,b).\) Then \(f\) has a directional derivative in the direction of the unit vector \(\vec{u}=u_1\vec{i}+u_2\vec{j}\) which is given by \[\begin{equation} \label{dirder} D_{\vec{u}}f(a,b)=f_x(a,b)u_1+f_y(a,b)u_2. \end{equation}\] :::

::: {.proof } We define a function \(F\) of a single variable \(h\) by \(F(h)=f\left(a+h u_1,b+h u_2\right)\) so that \[ D_{\vec{u}}f(a,b)=\lim _{h\to 0}\frac{f\left(a+h u_1,b+h u_2\right)-f(a,b)}{h} =\lim _{h\to 0}\frac{F(h)-F(0)}{h} =F'(0). \]
Applying the chain rule with \(x=a+h u_1\) and \(y=b+h u_2\) \[ F'(h) =\frac{d F}{d h} =\frac{\partial f}{\partial x}\frac{d x}{d h} +\frac{\partial f}{\partial y}\frac{d y}{d h} =f_x(x,y) u_1+f_y(x,y)u_2.\] When \(h=0,\) we have \(x=a\) and \(y=b\) so that \[ D_{\vec{u}} f(a,b)=F'(0)=\frac{\partial f}{\partial x}u_1 +\frac{\partial f}{\partial y}u_2 =f_x(a,b) u_1+f_y(a,b)u_2. \] :::

Example 4.1 Find the directional derivative \(D_{\vec{u}}f(x,y)\) where \(f\) is the function defined by \[ f(x,y)=x^3-3x y+4y^2 \] and \(\vec{u}\) is the unit vector given by the angle \(\theta =\pi /6.\) What is \(D_{\vec{u}}f(1,2)?\)

Solution. We let \(\vec{u}=\cos \left(\frac{\pi }{6}\right)\vec{i}+\sin \left(\frac{\pi }{6}\right)\vec{j}\) and use \(\ref{dirderivative}\) to find \[\begin{align*} D_{\vec{u}}f(x,y) & =f_x(x,y)\cos \frac{\pi }{6}+f_y(x,y)\sin \frac{\pi }{6} \\ & = \left(3x^2-3y\right)\frac{\sqrt{3}}{2}+(-3x+8y)\frac{1}{2} =\frac{1}{2}\left[3\sqrt{3}x^2-3x+\left(8-3\sqrt{3}\right)y\right].\end{align*}\] Therefore, by \(\ref{dirderivative}\), the directional derivative is \[ D_{\vec{u}}f(1,2) =\frac{1}{2}\left[3\sqrt{3}(1)^2-3(1)+8-3\sqrt{3}\right](2) =\frac{13-3\sqrt{3}}{2}. \]

Example 4.2 Find the directional derivative of the function \(f\) defined by
\[ f(x,y)=y^2+3y x^2 \] at \(P=(-1,-2)\) in the direction towards the origin.

Solution. A vector in the direction from \((-1,-2)\) to \((0,0)\) is \(\vec{i}+2\vec{j},\) so a unit vector in this direction is therefore \[ \vec{u}=\frac{1}{\sqrt{5}}\vec{i}+\frac{2}{\sqrt{5}}\vec{j}. \] We find that \[ f_y(-1,-2)=\left.\left(2y+3x^2\right)\right|_{(-1,-2)}=-1 \qquad \text{and} \qquad f_x(-1,-2)=\left.6x y\right|_{(-1,-2)}=12. \] Therefore, by \(\ref{dirderivative}\), the directional derivative is \[ D_{\vec{u}}f(-1,-2)=\frac{1}{\sqrt{5}}(12)+\frac{2}{\sqrt{5}}(-1)=2 \sqrt{5}. \]

Example 4.3 Find the directional derivative of the function \(f\) defined by
\[ f(x,y)=x^2+3x y^2 \] at \(P=(1,2)\) in the direction towards the origin.

Solution. A vector in the direction from \((1,2)\) to \((0,0)\) is \(-\vec{i}-2\vec{j},\) so a unit vector in this direction is therefore \[ \vec{u}=\frac{-1}{\sqrt{5}}\vec{i}-\frac{2}{\sqrt{5}}\vec{j}. \] We find that \[ f_x(1,2)=\left.\left(2x+3y^2\right)\right|_{(1,2)}=14 \qquad \text{and} \qquad f_y(1,2)=\left.6x y\right|_{(1,2)}=12. \] Therefore, the directional derivative is given by \[ D_{\vec{u}}f(1,2)=\frac{-1}{\sqrt{5}}(14)-\frac{2}{\sqrt{5}}(12)=\frac{-38}{\sqrt{5}}. \]

4.2 The Gradient of a Function

Definition 4.2 Let \(f\) be a differentiable function at \((a,b)\). Then the gradient of \(f\) is denoted by \(\nabla f(x,y)=f_x(x,y)\vec{i}+f_y(x,y)\vec{j}.\)
The value of the gradient at the point \(P(a,b)\) is denoted by \[ \nabla f_P=f_x(a,b)\vec{i}+f_y(a,b)\vec{j}. \]

Example 4.4 Find the gradient \(\nabla f\) of the function \(f\) defined by \[ f(x,y)=\sin x+e^{x y} \] and evaluate the gradient at \((0,1)\).

Solution. If \(f(x,y)=\sin x+e^{x y},\) then \[ \nabla f(x,y)=\langle \cos x+y e^{x y},x e^{x y}\rangle \] and \(\nabla f(0,1)=\langle 2,0 \rangle.\)

::: {#thm- } [Directional Derivative and Gradient] If \(f\) is a differentiable function of \(x\) and \(y\), then the directional derivative of \(f\) at the point \(P(a,b)\) in the direction of the unit vector \(\vec{u}\) is \[ D_{\vec{u}}f(a,b)=\nabla f_P\cdot \vec{u}. \] :::

Proof. Since \(\nabla f_P=f_x(a,b)\vec{i}+f_y(a,b)\vec{j}\) and \(\vec{u}=u_1\vec{i}+u_2\vec{j},\) we have \[ \nabla f_P\cdot \vec{u}=f_x(a,b)u_1+f_y(a,b)u_2=D_{\vec{u}}f(a,b). \]

Example 4.5 Find the directional derivative of the function \(f\) defined by \[ f(x,y)=x^2+x y+y^2 \] at \(P(1,-1)\) in the direction towards the origin.

Solution. We use the unit vector
\[ \vec{u}=\left(-\frac{1}{\sqrt{2}}\right)\vec{i}+\left(\frac{1}{\sqrt{2}}\right)\vec{j}\] and find
\(\nabla f=(2x+y)\vec{i}+(2y+x)\vec{j}\)
to determine \[ D_{\vec{u}}f(1,-1)=\nabla f(1,-1)\cdot \vec{u} =(\vec{i}-\vec{j})\cdot \left(-\frac{1}{\sqrt{2}}\right) \vec{i}+\left(\frac{1}{\sqrt{2}}\right)\vec{j}=-\sqrt{2}. \]

Example 4.6 Find the directional derivative of the function \(f\) defined by \[ f(x,y)=x^3y^4 \] at the point \((6,-1)\) in the direction of the vector \(\vec{v}=2\vec{i}+5\vec{j}.\)

Solution. We first compute the gradient vector at \((6,-1)\) \[ \nabla f(x,y)=3x^2y^4\vec{i}+4x^3y^3\vec{j} \qquad \text{and} \qquad \nabla f(6,-1)=108\vec{i}-864\vec{j}. \] Note that \(\vec{v}\) is not a unit vector, but since \(|\vec{v}|=\sqrt{29}\), the unit vector in the direction of \(\vec{v}\) is \[ \vec{u}=\frac{\vec{v}}{|\vec{v}|}=\frac{2}{\sqrt{29}}\vec{i}+\frac{5}{\sqrt{29}}\vec{j}. \] Therefore, by \(\ref{grader}\), we have \[ D_{\vec{u}}f(6,-1) =\nabla f(6,-1)\cdot \vec{u} =(108\vec{i}-864\vec{j})\cdot \left( \frac{2}{\sqrt{29}}\vec{i}+\frac{5}{\sqrt{29}}\vec{j}\right) =\frac{-4104}{\sqrt{29}}. \]

Example 4.7 Let \(f(x,y,z)=x y z,\) and let \(\vec{u}\) be a unit vector perpendicular to both \(\vec{v}=\vec{i}-2\vec{j}+3\vec{k}\) and \(\vec{w}=2 \vec{i}+\vec{j}-\vec{k}.\) Find the directional derivative of \(f\) at \(P(1,-1,2)\) in the direction \(\vec{u}.\)

Solution. The gradient is \[ \nabla f=\nabla (x y z)=y z \vec{i}+x z\vec{j}+x y \vec{k} \] and \(\nabla f_P=-2\vec{i}+2\vec{j}-\vec{k}.\) Now since we are looking for a unit vector perpendicular to both \(\vec{v}=\vec{i}-2\vec{j}+3\vec{k}\) and \(\vec{w}=2 \vec{i}+\vec{j}-\vec{k}\) we find, \(\vec{v} \times \vec{w}=-\vec{i}+7\vec{j}+5\vec{k}\) and so \(\vec{u}=\frac{1}{\sqrt{75}}(-\vec{i}+7\vec{j}+5\vec{k}).\) Therefore, by \(\ref{grader}\), we have \[ D_{\vec{u}}f =(-2\vec{i}+2\vec{j}-\vec{k})\cdot \left(\frac{-1}{\sqrt{75}}\vec{i}+\frac{7}{\sqrt{75}}\vec{j}+\frac{5}{\sqrt{75}}\vec{k}\right)=\frac{11\sqrt{3}}{15}. \]

::: {#thm- } [Properties of the Gradient] Let \(f\) and \(g\) be differentiable functions. Then

  • \(\nabla c=0\) for any constant \(c\)
  • \(\nabla (a f+b g)=a\nabla (f)+b\nabla (g)\) for any constants \(a\) and \(b\)
  • \(\nabla (f g)=f\cdot \nabla (g)+g\cdot \nabla (f)\)
  • \(\nabla \left(\frac{f}{g}\right)=\frac{g\cdot \nabla (f)-f\cdot \nabla (g)}{g^2}\) provided \(g^2\neq 0\)
  • \(\nabla \left(f^n\right)=n f^{n-1}\cdot \nabla (f)\)

:::

Proof. An outline of how each part can be proven in listed below.

  • The (constant rule) is proved as follows by \(\nabla c=(c)_x\vec{i}+(c)_y\vec{j}=0.\)
  • The (linearity rule) is proved as follows \[ \nabla (a f +b g) =\left(a f_x+ b g_x\right)\vec{i}+\left(a f_x+ b g_y\right)\vec{j} =a\nabla (f)+b\nabla (g). \]
  • The (product rule) is proved as follows \[\begin{align*} \nabla (f g) & =(f g)_x\vec{i}+(f g)_y\vec{j} =\left(f_xg+g_xf\right) \vec{i}+\left(f_yg+g_yf\right) \vec{j} \\ & =\left[f\left(g_x\vec{i}+g_y\vec{j}\right)\right]+\left[g\left(f_x\vec{i}+f_y\vec{j}\right)\right] =f\cdot \nabla (g)+g\cdot \nabla (f). \end{align*}\]
  • The (quotient rule) is proved as follows \[\begin{align*} \nabla \left(\frac{f}{g}\right) & =\left(\frac{f}{g}\right)_x\vec{i}+\left(\frac{f}{g}\right)_y\vec{j} =\left(\frac{f_xg-g_xf }{g^2}\right)\vec{i}+\left(\frac{f_yg-g_yf }{g^2}\right) \vec{j} \\ & =\frac{\left[g\left(f_x\vec{i}+f_y\vec{j}\right)\right]-\left[f\left(g_x\vec{i}+g_y\vec{j}\right)\right]}{g^2} =\frac{g\cdot \nabla (f)-f\cdot \nabla (g)}{g^2}. \end{align*}\]
  • The (power rule) is proved as follows \[ \nabla f^n =\left(f^n\right)_x\vec{i}+\left(f^n\right)_y\vec{j} =\left(n f^{n-1}\right)f_xb \vec{i}+\left(n f^{n-1}\right)f_yb \vec{j} =n f^{n-1}\cdot \nabla (f). \]

::: {#thm- } [Orthogonality of Gradient] Suppose the function \(f\) is differentiable at the point \(P\) and that the gradient at \(P\) satisfies \(\nabla f_P\neq 0.\) Then \(\nabla f_P\) is orthogonal to the level surface (or curve) of \(f\) through \(P\). :::

Proof. Let \(C\) be any smooth curve on the level surface \(f(x,y,z)=K\) that passes through \(P(a,b,c),\) and describe the curve \(C\) by the vector function \(\vec{R}(t)=x(t)\vec{i}+y(t)\vec{j}+z(t)\vec{k}\) for all \(t\) in some interval \(I\). We will show that the gradient \(\nabla f_P\) is orthogonal to the tangent vector \(d R/d t\) at \(P.\) Because \(C\) lies on the level surface, any point \(P(x(t),y(t),z(t))\) on \(C\) must satisfy \(f[x(t),y(t),z(t)]=K,\) and by applying the chain rule, we obtain \[\frac{d}{d t}[f(x(t),y(t),z(t))] =f_x(x,y,z)\frac{d x}{d t} +f_y(x,y,z)\frac{d y}{d t}+f_z(x,y,z)\frac{d z}{d t}. \] Suppose \(t=t_0\) at \(P.\) Then \[\begin{align*} & \left. \frac{d}{d t}\{ f[x(t),y(t),z(t)] \}\right|_{t=t_0} \\ & \qquad = f_x\left(x\left(t_0\right),y\left(t_0\right),z\left(t_0\right)\right)\frac{d x}{d t} + f_y\left(x\left(t_0\right),y\left(t_0\right),z\left(t_0\right)\right)\frac{d y}{d t} + f_z\left(x\left(t_0\right),y\left(t_0\right),z\left(t_0\right)\right)\frac{d z}{d t} \\ & \qquad = \nabla f_P\frac{d \vec{R}}{d t} \end{align*}\] since \[\frac{d \vec{R}}{d t}=\frac{d x}{d t}\vec{i}+\frac{d y}{d t}\vec{j}+\frac{d z}{d t}\vec{k}.\] We also know that \(f(x(t),y(t),z(t))=K\) for all \(t\) in \(I.\) Thus, we have \[\frac{d}{d t}\{f[x(t),y(t),z(t)] \}=\frac{d}{d t}(K)=0\] and it follows that \(\nabla f_P\cdot \frac{d \vec{R}}{d t}=0.\) We are given that \(\nabla f_P\neq 0\) and \(d \vec{R}/d t\neq 0\) because the curve \(C\) is smooth. Therefore, \(\nabla f_P\) is orthogonal to \(d \vec{R}/d t,\) as required.

4.3 Steepest Ascent and Steepest Descent

The direction of the greatest rate of increase (or decrease) of a given function at a specified point is called the direction of steepest ascent (or steepest descent).

::: {#thm- } [Steepest Ascent/Descent] Suppose \(f\) is differentiable at the point \(P\) and that the gradient of \(f\) at \(P\) satisfies \(\nabla f_P\neq 0.\) Then

  • The largest value of the directional derivative \(D_{\vec{u}}f\) at \(P\) is \(\left\|\nabla f_P\right\|\) and occurs when the unit vector \(\vec{u}\) points in the direction of \(\nabla f_P.\)
  • The smallest value of the directional derivative \(D_{\vec{u}}f\) at \(P\) is \(-\left\|\nabla f_P\right\|\) and occurs when the unit vector \(\vec{u}\) points in the direction of \(-\nabla f_P.\) :::

Proof. If \(\vec{u}\) is any unit vector, then\(D_{\vec{u}}f=\nabla f_P\cdot \vec{u}=\left\|\nabla f_P\right\|\|\vec{u}\| \cos \theta =\left\|\nabla f_P\right\| \cos \theta\) where \(\theta\) is the angle between \(\nabla f_P\) and \(\vec{u}.\) But \(\cos \theta\) assumes its largest value of 1 at \(\theta =0;\) that is, when \(\vec{u}\) points in the direction \(\nabla f_P.\) Thus, the largest possible value of \(D_{\vec{u}}f\) is \[D_{\vec{u}}f =\left\|\nabla f_P\right\|(1)=\left\|\nabla f_P\right\|.\] Also \(\cos \theta\) assumes its smallest value \(-1\) when \(\theta =\pi .\) This value occurs when \(\vec{u}\) points toward \(-\nabla f_P,\) and in this direction \[D_{\vec{u}}f =\left\|\left|\left.\nabla f_P\right\|(-1)\right.\right. =\left\|\left|\left.\nabla f_P\right\|(-1)\right.\right. =-\left\|\nabla f_P\right\|. \]

Example 4.8 Sketch the level curve corresponding to \(C=1\) for the function \[ f(x,y)=x^2-y^2 \] and find a normal vector at the point \(P(2,\sqrt{3}).\)

Solution. The level curve for \(C=1\) is a hyperbola given by \(x^2-y^2=1\). The gradient vector is perpendicular to the level curve. We have \[ \nabla f=f_x\vec{i}+f_y\vec{j}=2x\vec{i}-2y\vec{j} \] so at the point \(\left(2,\sqrt{3}\right),\) \(\nabla f_P=4i-2\sqrt{3}j\) is the required normal.

Example 4.9 In what direction is the function defined by \[ f(x,y)=x e^{2y-x} \] increasing most rapidly at the point \(P(2,1)\), and what is the maximum rate of increase? In what direction is \(f\) decreasing most rapidly?

Solution. We begin by finding the gradient of \(f,\) \[\begin{align*} \nabla f & =f_x\vec{i}+f_y\vec{j} \\ & =\left[e^{2y-x}+x e^{2y-x}(-1)\right]\vec{i}+\left[x e^{2y-x}(2)\right]\vec{j} \\ & =e^{2y-x}(1-x)\vec{i}+2 x\vec{j}\end{align*}\] At \(P(2,1)\), \[\nabla f_P=e^{2(1)-2}[(1-2)\vec{i}+2(2)\vec{j}]=-\vec{i}+4\vec{j}.\]
The most rapid rate of increase is \(\left\|\nabla f_P\right\|=\sqrt{(-1)^2+4^2}=\sqrt{17}\) and occurs in the direction of \(-\vec{i}+4\vec{j}.\) The most rapid rate of decrease occurs in the direction of \(-\nabla f_P=\vec{i}-4\vec{j}\) and is \(-\sqrt{17}.\)

Example 4.10 Let \(f(x,y,z)=y e^{x+z}+z e^{y-x}.\) At the point \(P(2,2,-2),\) find the unit vector pointing in the direction of most rapid increase of \(f.\)

Solution. The gradient is \[ \nabla f=\left(y e^{x+z}-z e^{y-x}\right)\vec{i}+\left(e^{x+z}+x e^{y-x}\right)\vec{j}+\left(y e^{x+z}+e^{y-z}\right)\vec{k}. \] Thus, \(\nabla f(2,2,-2)=4\vec{i}-\vec{j}+3\vec{k}\) and so \(\vec{u}=\frac{1}{\sqrt{26}}(4\vec{i}-\vec{j}+3\vec{k}).\)

Example 4.11 Find the maximum rate of change of the function at the given point and the direction in which it occurs \(f(x,y)=\sqrt{x^2+2y}\), \((4,10)\)

Solution. We find \[ \nabla f(4,10)=\left.\frac{x}{\sqrt{x^2+2 y}}\right|_{(4,10)}\vec{i}+\left.\frac{1}{\sqrt{x^2+2 y}}\right|_{(4,10)}\vec{j} =\frac{2}{3}\vec{i}+\frac{1}{6}\vec{j}. \]
Thus the maximum rate of change is \(\sqrt{(2/3)^2+(1/6)^2}=\frac{\sqrt{17}}{6}\) and occurs in the direction of \(\frac{2}{3}\vec{i}+\frac{1}{6}\vec{j}.\)

Example 4.12 Find the maximum rate of change of the function at the given point and the direction in which it occurs \(f(x,y)=\sqrt{y^2+2x}\), \((-4,-10)\)

Solution. We find \[ \nabla f(-4,-10)=\left.\frac{1}{\sqrt{2 x+y^2}}\right|_{(-4,-10)}\vec{i}+\left.\frac{y}{\sqrt{2 x+y^2}}\right|_{(-4,-10)}\vec{j} =\frac{1}{2 \sqrt{23}}\vec{i}-\frac{5}{\sqrt{23}}\vec{j}. \]
Thus the maximum rate of change is \[ \sqrt{\left(\frac{1}{2 \sqrt{23}}\right)^2+\left(-\frac{5}{\sqrt{23}}\right)^2}=\frac{1}{2}\sqrt{\frac{101}{23}} \]
and occurs in the direction of \(\frac{1}{2 \sqrt{23}}\vec{i}-\frac{5}{\sqrt{23}}j\vec{.}\)

Example 4.13 Find the direction from \(P(2,-1,2)\) in which the function
\(f(x,y,z)=(x+y)^2+(y+z)^2+(x+z)^2\) increases most rapidly and compute the magnitude of the greatest rate of increase.

Solution. We compute the gradient:
\[ \nabla f=(4 x+2y+2z)\vec{ i}+(2x+4y+2z)\vec{ j}+(2x+2y+4z)\vec{ k} \] and at \((2,-1,2)\) we have,
\(\nabla f(2,-1,2)=10\vec{ i}+4\vec{ j}+10\vec{ k}\). Therefore, \(\|\nabla f\|=\sqrt{216}\) is the magnitude of the greatest rate of increase and occurs in the direction of \(10\vec{ i}+4\vec{ j}+10\vec{ k}.\)

4.4 Tangent Planes

Suppose \(S\) is a surface with the equation \(z=f(x,y)\) where \(f\) has continuous first partial derivatives \(f_x\) and \(f_y\). Let \(P(a,b,c)\) be a point on \(S\) and let \(C_1\)be the curve of intersection of \(S\) with the plane \(x=a\) and \(C_2\) the intersection of \(S\) with the plane \(y=b\). The tangent lines \(T_1\) and \(T_2\) to \(C_1\) and \(C_2\), respectively determine a unique plane and this plane actually contains the tangent to every smooth curve \(C\) that passes through \(P.\) We call this plane the tangent plane to \(S\) at \(P\).

::: {#thm- } [Equation of Tangent Plane] If \(z=f(x,y)\) is differentiable at \((a,b)\) then an equation of the tangent plane to the graph of \(f\) at \((a,b)\) is \[ z-f(a,b)=f_x(a,b)(x-a)+f_y(a,b)(y-b). \] :::

Proof. The proof is left for the reader.

Example 4.14 Find an equation of the tangent plane to the surface \[ z=2x^2+y^2 \] at the point \((1,1,3)\).

Solution. Let \(f(x,y)=2x^2+y^2.\) Then \[ f_x(x,y)=4x, \quad f_x(1,1)=4, \quad f_y(x,y)=2y, \quad f_y(1,1)=2. \] Then an equation of the tangent plane is \(z-3=4(x-1)+2(y-1)\) or \[\begin{equation} z=4x+2y-3. \tag*{} \end{equation}\]

Example 4.15 Find an equation of the tangent plane to the surface \[ z=4-x^2-y^2 \] at the point \((1,1,2)\).

Solution. Let \(f(x,y)=4-x^2-y^2.\) Then \[ f_x(x,y)=-2x, \quad f_x(1,1)=-2, \quad f_y(x,y)=-2y, \quad f_y(1,1)=-2. \] Then an equation of the tangent plane is \(z-2=-2(x-1)-2(y-1)\) or \[\begin{equation} z=-2x-2y+6 \tag*{} \end{equation}\]

Example 4.16 Find an equation of the tangent plane to the surface \[ z=\sin x+\sin y \] at the point \(\left(\frac{\pi}{2},\frac{\pi}{3},1+\frac{\sqrt{3}}{2}\right)\).

Solution. Let \(f(x,y)=\sin x+\sin y\). Then \[ f_x(x,y)=\cos x, \quad f_x\left(\frac{\pi}{2},\frac{\pi}{3}\right)=0, \quad f_y(x,y)=\cos y, \quad f_y\left(\frac{\pi}{2},\frac{\pi}{3}\right)=\frac{1}{2}. \] Then an equation of the tangent plane is \[ z-\left(1+\frac{\sqrt{3}}{2}\right)=0\left(x-\frac{\pi}{2}\right)+\frac{1}{2}\left(y-\frac{\pi}{3}\right) \] or \[\begin{equation} z=\frac{1}{2}y-\frac{\pi}{6}+1+\frac{\sqrt{3}}{2} \tag*{} \end{equation}\]

::: {#thm- } [Tangent Plane To A Surface] Suppose \(S\) is a surface with the equation \(F(x,y,z)=C\) and let \(P(a,b,c)\) be a point on \(S\) where \(F\) is differentiable with \(\nabla F_P\neq 0.\) Then an equation of the tangent plane to \(S\) at \(P\) is \[ F_x(a,b,c)(x-a)+F_y(a,b,c)(y-b)+F_z(a,b,c)(z-c)=0 \]
and the normal line to \(S\) at \(P\) has parametric equations \[ x=a+F_x(a,b,c) t, \qquad y=b+F_y(a,b,c) t, \qquad z=c+F_z(a,b,c) t.\] :::

Proof. Any plane that passes through \(P(a,b,c)\) has an equation of the form \[ A(x-a)+B(y-b)+C(z-c)=0. \] By dividing this equation by \(C\) and letting \(a_1=-A/C\) and \(b_1=-B/C\), we can write it in the form \[ z-c=a_1(x-a)+b_1(y-b). \] If this equation represents the tangent plane at \(P\), then its intersection with the plane \(y=b\) must be the tangent line with slope \(f_x(a,b)\). Therefore, \(a_1=f_x(a,b).\) Similarly, putting \(x=a\), we get \(z-c=b_2(y-b),\) which must represent the tangent line with slope \(f_y(a,b)\) and so \(b_2=f_y(a,b).\) Thus, the equation of the tangent plane is \(z-c=f_x(a,b)(x-a)+f_y(a,b)(y-b).\)

Example 4.17 Find an equation of the tangent plane to the surface \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 \] at the point \((x_0,y_0,z_0)\).

Solution. Since \(F_x=2x/a^2,\) \(F_y=2y/b^2,\) and \(F_z=2z/c^2,\) the tangent plane at \(P_0\left(x_0,y_0,z_0\right)\) has equation
\[ \frac{2x_0}{a^2}\left(x-x_0\right)+\frac{2y_0}{b^2}\left(y-y_0\right)+\frac{2z_0}{c^2}\left(z-z_0\right)=0\] \[ \frac{x_0 x}{a^2}-\frac{x_0{}^2}{a^2}+\frac{y_0y}{b^2}-\frac{y_0^2}{b^2}+\frac{z_0z}{c^2}-\frac{z_0{}^2}{c^2}=0\] \[ \frac{x_0 x}{a^2}+\frac{y_0y}{b^2}+\frac{z_0z}{c^2}=1. \]

Example 4.18 Find an equation of the tangent plane to the surface \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1 \] at the point \((x_0,y_0,z_0)\).

Solution. Since \(F_x=2x/a^2,\) \(F_y=2y/b^2,\) and \(F_z=-2z/c^2,\) the tangent plane at \(P_0\left(x_0,y_0,z_0\right)\) has equation
\[ \frac{2x_0}{a^2}\left(x-x_0\right)+\frac{2y_0}{b^2}\left(y-y_0\right)-\frac{2z_0}{c^2}\left(z-z_0\right)=0\] \[ \frac{x_0 x}{a^2}-\frac{x_0^2}{a^2}+\frac{y_0y}{b^2}-\frac{y_0^2}{b^2}+\frac{z_0z}{c^2}+\frac{z_0^2}{c^2}=0\] \[ \frac{x_0 x}{a^2}+\frac{y_0y}{b^2}-\frac{z_0z}{c^2}=1. \]

Example 4.19 Find an equation of the tangent plane to the surface \[ \frac{z}{c}=\frac{x^2}{a^2}+\frac{y^2}{b^2} \] at the point \((x_0,y_0,z_0)\).

Solution. Since \(F_x=2x/a^2,\) \(F_y=2y/b^2,\) and \(F_z=-1/c,\) the tangent plane at \(P_0\left(x_0,y_0,z_0\right)\) has equation \[ \frac{2x_0}{a^2}\left(x-x_0\right)+\frac{2y_0}{b^2}\left(y-y_0\right)-\frac{1}{c}\left(z-z_0\right)=0\] \[ \frac{x_0 x}{a^2}-\frac{x_0^2}{a^2}+\frac{y_0y}{b^2}-\frac{y_0{}^2}{b^2}+\frac{z}{c}+\frac{z_0}{c}=0 \] \[ \frac{x_0 x}{a^2}+\frac{y_0y}{b^2}-\frac{z}{c}=\frac{z_0}{c}. \]

4.5 Normal Lines

Example 4.20 Find the equations for the tangent plane and the normal line to the cone \[ z^2=x^2+y^2 \] at the point where \(x=3, y=4,\) and \(z>0.\)

Solution. If \(P(a,b,c)\) is the point of tangency and \(a=3, y=4,\) and \(c>0,\) then \(c=\sqrt{a^2+b^2}=5.\) If we consider \(F(x,y,z)=x^2+y^2-z^2,\) then the cone can be regarded as the level surface \(F(x,y,z)=0.\) The partial derivatives of \(F\) are \(F_x=2x\), \(F_y=2y\), and \(F_z=-2z\) so at \(P(3,4,5)\) we find \(F_x(3,4,5)=6\), \(F_y(3,4,5)=8,\) and \(F_z(3,4,5)=-10.\) Thus the tangent plane has an equation \[6(x-3)+8(y-4)-10(z-5)=0\] or \(3x+4y-5z=0\) and the normal line is given parametrically by the equations \(x=3+6t ,\) \(y=4+8t,\) and \(z=5-10t.\)

Example 4.21 Find the equations of the tangent plane and the normal line at the point \((-2,1,-3)\) to the ellipsoid \[ \frac{x^2}{4}+y^2+\frac{z^2}{9}=3. \]

Solution. The ellipsoid is a level surface of the function \[ F(x,y,z)=\frac{x^2}{4}+y^2+\frac{z^2}{9}. \] Therefore we have
\[ F_x(x,y,z)=\frac{x}{2}, \quad F_y(x,y,z)=2y, \quad F_z(x,y,z)=\frac{2z}{9} \] Thus, \[ F_x(-2,1,3)=-1, \quad F_y(-2,1,3)=2, \quad F_z(-2,1,3)=-2/3. \]
Then the tangent plane at \((-2,1,-3)\) is \[ -1(x+2)+2(y-1)-(2/3)(z+3)=0 \] which simplifies to \[ 3x-6y+2z+18=0 \] and parametric equations for the normal line are \[ x=-2- t, \quad y=1+2t, \quad z=-3-(2/3)t. \]

4.6 Exercises

Exercise 4.1 Find the directional derivative of \(f(x,y)=x^2y^2-x^2+2y\) at the point \((2,2)\) in the direction of the unit vector \(\vec{u}=\frac{1}{2}\vec{i}-\frac{\sqrt{3}}{2}\vec{j}\).

Exercise 4.2 Find the gradient of the function \(f(x,y)=y \tan x+\sin xy\).

Exercise 4.3 Find the directional directive of \(f(x,y)=\frac{e^{-x}}y\) at \(P(0,-1)\) in the direction of \(\mathbf{v}=-\mathbf{i}+\mathbf{j}\).

Exercise 4.4 Find the gradient of the function.

  • \(f(x,y)=x^2-2 x y\)
  • \(f(x,y)=\ln \left(x^2+y^2\right)\)
  • \(f(x,y)=\sin (x+2y)\)
  • \(f(x,y,z)=\frac{x y-1}{z+x}\)
  • \(f(x,y,z)= x y z^2\)
  • \(g(x,y,z)=x e^{y+3z}\)

Exercise 4.5 Compute the directional derivative of the function \(f(x,y)=\ln \left(x^2+3y\right)\) at the point \((1,1)\) in the direction of the vector \(v=\vec{i}+\vec{j}.\)

Exercise 4.6 Compute the directional derivative of the function \(f(x,y)=\sin x y\) at the point \(\left(\sqrt{\pi },\sqrt{\pi }\right)\) in the direction of the vector \(v=3\pi \vec{i}-\pi \vec{j} .\)

Exercise 4.7 Find the directional derivative of the given function at the given point in the direction of the given vector.

  • \(f(x,y)=x^2+x y\), \((1,-2),\) and \(\vec{i}+\vec{j}\)
  • \(f(x,y)=\frac{e^{-x}}{y},\) \((2,-1),\) and \(-\vec{i}+\vec{j}\)

Exercise 4.8 Find the directional derivative of the given function at the given point in the direction of the given vector.

  • \(f(x,y)=\ln \left(3x+y^2\right)\), \((0,1),\) and \(\vec{i}-\vec{j}\)
  • \(f(x,y)=\sec \left(x y-y^3\right),\) \((2,0),\) and \(-\vec{i}-3\vec{j}\)

Exercise 4.9 Find the direction from \(P_0\) in which the given function \(f\) increases most rapidly and compute the magnitude of the greatest rate of increase.

  • \(f(x,y,z)=(x+y)^2+(y+z)^2+(x+z)^2\) at \(P_0(2,-1,2)\)
  • \(f(x,y,z)=z \ln \left(\frac{y}{z}\right)\) at \(P_0(1,e,-1)\)

4.7 Extreme Values of Two Variable Functions

4.8 Relative Extrema

Definition 4.3 Let \(f\) be a function of two variables \(x\) and \(y\).

  • The function \(f\) has a relative maximum at \(\left(x_0,y_0\right)\) if \(f(x,y)\leq f\left(x_0,y_0\right)\) for all \((x,y)\) in an open disk containing \(\left(x_0,y_0\right)\).
  • The function \(f\) has a relative minimum at \(\left(x_0,y_0\right)\) if \(f(x,y)\geq f\left(x_0,y_0\right)\) for all \((x,y)\) in an open disk containing \(\left(x_0,y_0\right).\)

Collectively, relative maxima and relative minima are called relative extrema .

::: {#lem- } [Critical Points]

If \(f\) has a relative extremum at \(\left(x_0,y_0\right)\) and partial derivatives \(f_x\) and \(f_y\) both exist at \(\left(x_0,y_0\right),\) then \[ f_x\left(x_0,y_0\right)=f_y\left(x_0,y_0\right)=0. \] :::

Proof. Let \(F(x)=f\left(x,y_0\right).\) Then \(F(x)\) must have a relative extremum at \(x=x_0,\) so \(F'\left(x_0\right)=0,\) which means that \(f_x\left(x_0,y_0\right)=0.\) Similarly, \(G(y)=f\left(x_0,y\right)\) has a relative extremum at \(y=y_0,\) so \(G'\left(y_0\right)=0\) and \(f_y\left(x_0,y_0\right)=0.\) Thus, we must have both \(f_x\left(x_0,y_0\right)=0\) and \(f_y\left(x_0,y_0\right)=0.\)

Definition 4.4 A critical point of a function \(f\) defined on an open set \(D\) is a point \(\left(x_0,y_0\right)\) in \(D\) where either one of the following is true:

  • \(f_x\left(x_0,y_0\right)=f_y\left(x_0,y_0\right)=0\) or
  • at least one of \(f_x\) or \(f_y\) does not exist at \(\left(x_0,y_0\right).\)

Example 4.22 Find the critical points for the function \[ f(x,y)=x^2+y^2-2x-6y+14. \]

Solution. The first partial derivatives of \(f\) are \[ f_x(x,y)=2x-2 \qquad \text{and} \qquad f_y(x,y)=2y-6. \] These partial derivatives are equal to 0 when \(x=1\) and \(y=3\) so the only critical point is \((1,3).\) By completing the square we find that \[ f(x,y)=4+(x-1)^2+(y-3)^2. \]
Since \((x-1)^2\geq 0\) and \((y-3)^2\geq 0,\) we have \(f(x,y)\geq 4\) for all values of \(x\) and \(y.\) Therefore, \(f(1,3)=4\) is a local minimum, and in fact it is the absolute minimum of \(f.\) This can be confirmed geometrically from the graph of \(f\), which is the elliptic paraboloid with vertex \((1,3,4)\) as shown in \(\ref{fig:ext-func-1}\).

Definition 4.5 A critical point \(P_0\left(x_0,y_0\right)\) is called a saddle point of \(f\) if every open disk centered at \(P_0\) contains points in the domain of \(f\) that satisfy \(f(x,y)>f\left(x_0,y_0\right)\) as well as points in the domain of \(f\) that satisfy \(f(x,y)<f\left(x_0,y_0\right).\)

Example 4.23 Find the extreme values of \[ f(x,y)=y^2-x^2. \]

Solution. Since \(f_x=-2x\) and \(f_y=2y,\) the only critical point is \((0,0).\) Notice that for points on the \(x\)-axis we have \(y=0,\) so \(f(x,0)=-x^2<0\) (if \(x\neq 0.\)) However for points on the \(y\)-axis we have \(x=0,\) so \(f(0,y)=y^2>0\) (if \(y\neq 0\)). Thus every disk with center \((0,0)\) contains points where \(f\) takes positive values as well as points where \(f\) takes negative values. Therefore \(f(0,0)=0\) cannot be an extreme value for \(f,\) so \(f\) has no extreme values.

This example illustrates the fact that a function need not have a maximum or minimum value at a critical point. The graph of \(f\) is the hyperbolic paraboloid which has a horizontal tangent plane \((z=0)\) at the origin. You can see that \(f(0,0)=0\) is a maximum in the direction of the \(x-\text{axis}\) but not in the direction of the \(y\)-axis. Near the origin the graph has the shape of a saddle as shown in \(\ref{fig:ext-func-2}\).

::: {#thm- } [Second Partials Test] Assume \(f\) has a critical point at \(P_0(x_0,y_0)\) and assume that \(f\) has continuous second order partial derivatives in a disk centered at \(\left(x_0,y_0\right).\) If \(D:=(f_{x x}f_{y y}-\left(f_{x y}\right)^2)\left(x_0,y_0\right)>0\), then

  • a relative maximum occurs at \(P_0\) if \(f_{x x}\left(x_0,y_0\right)<0\)
  • a relative minimum occurs at \(P_0\) if \(f_{x x}\left(x_0,y_0\right)>0\).

If \(D\left(x_0,y_0\right)<0\), then a saddle point occurs at \(P_0\). :::

Example 4.24 Find the relative extrema of the function
\[ f(x,y)=x^4+y^4-4x y+1. \]

Solution. We first locate the critical points \[ f_x=4x^3-4y \qquad \text{and} \qquad f_y=4y^3-4x. \]
Setting these partial derivatives to 0, we obtain the equations\(x^3-y=0\) and \(y^3-x=0.\)
To solve these equations we substitute \(y=x^3\) from the first equation into the second one. This gives \[ 0 =x^9-x =x\left(x^8-1\right) =x\left(x^4-1\right)\left(x^4+1\right) =x(x-1)(x+1)\left(x^2+1\right)\left(x^4+1\right) \] So there are three real roots: \(x=0,1 ,-1.\) The three critical points are \((0,0), (1,1),\) and \((-1,-1).\) Next we calculate the second partial derivatives and \(D(x,y)\): \[ f_{x x}=12x^2, \qquad f_{x y}=-4, \qquad f_{y y}=12y^2. \] Thus, \[D(x,y)=f_{x x}f_{y y}-\left(f_{x y}\right)^2=144x^2y^2-16. \] Since \(D(0,0)=-16<0,\) it follows that the origin is a saddle point; that is, \(f\) has no local extremum at \((0,0).\) Since \(D(1,1)=128>0\) and \(f_{x x}(1,1)=12>0,\) we see that \(f(1,1)=-1\) is a local minimum. Similarly, we have \(D(-1,-1)=128>0\) and \(f_{x x}(-1,1-1)=12>0,\) so \(f(-1,-1)=-1\) is also a local minimum.

4.9 Absolute Extrema

Definition 4.6 Let \(f\) be a function of two variables \(x\) and \(y\).

  • The function \(f\) has a relative maximum at \(\left(x_0,y_0\right)\) if \(f(x,y)\leq f\left(x_0,y_0\right)\) for all \((x,y)\) in the domain \(D\) of \(f\).
  • The function \(f\) has a relative minimum at \(\left(x_0,y_0\right)\) if \(f(x,y)\geq f\left(x_0,y_0\right)\) for all \((x,y)\) in the domain \(D\) of \(f\).

Collectively, absolute maxima and absolute minima are called absolute extrema .

::: {#thm- } [Extreme Value] A function \(f\) attains both an absolute maximum and an absolute minimum on any closed bounded set \(S\) where it is continuous. :::

Proof. The proof is left for the reader.

Example 4.25 Find the shortest distance from the point \((1,0,-2)\) to the plane \[ x+2y+z=4. \]

Solution. The distance from any point \((x,y,z)\) to the point \((1,0,-2)\) is \[d=\sqrt{(x-1)^2+y^2+(z+2)^2}\] but if \((x,y,z)\) lies on the plane, then \(z=4-x-2y\) and so we have \[d=\sqrt{(x-1)^2+y^2+(6-x-2y)^2}.\] We can minimize \(d\) by minimizing the simpler expression \[d^2=f(x,y)=(x-1)^2+y^2+(6-x-2y)^2.\]
By solving the equations \[f_x=2(x-1)^2-2(6-x-2y)=4x+4y-14=0\]
\[f_y=2y-4(6-x-2y)=4x+10y-24=0\] we find that the only critical point is \((11/3,5/3).\) Since \(f_{x x}=4,\) \(f_{x y}=4,\) and \(f_{y y}=10,\) we have \[D(x,y)=f_{x x}f_{y y}-\left(f_{x y}\right){}^2=24>0\] and \(f_{x x}>0\) so \(f\) has a local minimum at \((11/6,5/3).\) Intuitively, we can see that this local minimum is actually an absolute minimum because there must be a point on the given plane that is closest to \((1,0,-2).\) If \(x=11/6\) and \(y=5/3,\) then \[d=\sqrt{\left(\frac{5}{6}\right)^2+\left(\frac{5}{3}\right)^2+\left(\frac{5}{6}\right)^2}=\frac{5\sqrt{6}}{6}.\] which is the shortest distance.

Example 4.26 Find the absolute extrema of the function over the bounded region \[ f(x,y)=x^2-2x y+2y \] over the rectangle \[ D=\{(x,y) \mid 0\leq x\leq 3,0\leq y\leq 2\}. \]

Solution. Since \(f\) is a polynomial it is continuous on the closed bounded rectangle \(D,\) therefore \(f\) has both absolute maximum and minimum values. We first find the critical points by solving the system \[f_x=2x-2y=0 \qquad \text{and} \qquad f_y=-2x+2=0\] The only critical point is \((1,1),\) and the value of \(f\) there is \(f(1,1)=1.\) We look at the values of \(f\) on the boundary of \(D\), which consists of four line segments \(L_1, L_2, L_3\), and \(L_4\) as follows. On \(L_1\) we have \(y=0\) and \(f(x,0)=x^2\) for \(0\leq x\leq 3.\) This is an increasing function of \(x,\) so its minimum value is \(f(0,0)=0\) and its maximum value is \(f(3,0)=9.\) On \(L_2\) we have \(x=3\) and \(f(3,y)=9-4y\) on \(0\leq y\leq 2.\) This is a decreasing function of \(y,\) so its maximum value is \(f(3,0)=9\) and its minimum value is \(f(3,2)=1.\) On \(L_3\) we have \(y=2\) and \(f(x,2)=x^2-4x+4\) on \(0\leq x\leq 3.\) By observing that \(f(x,2)=(x-2)^2,\) we see that the minimum value of this function is \(f(2,2)=0\) and the maximum value is \(f(0,2)=4.\) On \(L_4\) we have \(x=0\) and \(f(0,y)=2y\) on \(0\leq y\leq 2\) with maximum value \(f(0,2)=4\) and minimum value \(f(0,0)=0.\)

Thus on the boundary, the minimum value of \(f\) is 0 and the maximum is 9. We compare these values with the value \(f(1,1)=1\) at the critical point and conclude that the absolute maximum value of \(f\) on \(D\) is \(f(3,0)=9\) and the absolute minimum value is \(f(0,0)=f(2,2)=0.\)

Example 4.27 A rectangular box without a lid is to be made from 12 \(m^2\) of cardboard. Find the maximum volume of such a box.

Solution. Let the length, width, and height of the box (in meters) be \(x,\) \(y,\) and \(z.\) Then the volume of the box is \(V=x y z.\) We can express \(V\) as a function of just two variables by using the fact that the surface area of the sides and the bottom of the box is \(2x z+2 y z+x y=12.\) Solving these equation for \(z,\) we get \[ z=\frac{12-x y}{2(x+y)}, \] so the expression for volume \(V\) becomes \[ V=\frac{12 x y-x^2y^2}{2(x+y)}. \] We compute the partial derivatives \[ V_x=\frac{y^2\left(12-2 x y-x^2\right)}{2(x+y)} \qquad \text{and}\qquad V_y=\frac{x^2\left(12-2 x y -y^2\right)}{2(x+y)^2}. \] If \(V\) is a maximum, then \(V_x=V_y=0,\) but \(x=0\) or \(y=0\) gives \(V=0,\) so we must solve the equations \[ 12-2x y-x^2=0\qquad \text{and} \qquad 12-2 x y-y^2=0. \] These imply that \(x^2=y^2\) and so \(x=y\). (Note that \(x\) and \(y\) must both be positive in this example.) If we put \(x=y\) in either equation we get \(12-3x^2=0,\) which gives \(x=2,\) \(y=2,\) and \(z=1.\) From the physical nature of this example there must be an absolute maximum volume that has to occur at a critical point of \(V,\) so it must be when \(x=2,\) \(y=2,\) and \(z=1.\) Then \(V=2\cdot 2\cdot 1=4,\) so the maximum volume of the box is \(4m^3.\)

Example 4.28 Find the absolute extrema of the function \[ f(x,y)=x^2+3y^2-4x+2y-3 \] over the rectangle \[ D=\{(x,y) \mid 0\leq x\leq 3,-3\leq y\leq 0\}. \]

Solution. We compute \(f_x=2x-4\) and \(f_y=6y+2\) and set \(f_x=f_y=0\) and find that \((2,-1/3)\) is the only critical point in the interior. On \(x=t, y=0\) for \(0\leq t\leq 3\), we have \[ F_1(t)=f(t,0)=t^2-4t-3=0 \] and so \(F_1'(t)=2t-4\) yielding the point \((2,0).\) On \(x=3, y=t\) for \(-3\leq t\leq 0\), we have \[ F_2(t)=f(3,t)=3t^2+2t-6 \] and so \(F_2'(t)=6t+2=0\) yielding the point \((3,-1/3).\) On \(x=t, y=-3\) for \(0\leq t\leq 3\), we have
\[ F_3(t)=f(t,-3)=t^2-4t+18 \] and so \(F_3'(t)=2t-4=0\) yielding the point \((2,-3).\) On \(x=0, y=t\) for \(-3\leq t\leq 0\), we have \[ F_4'(t)=f(0,t)=3t^2+2t-3 \] and so \(F_4'(t)=6t+2=0\) yielding the point \((0,-1/3).\)

Finally, we have \(f(2,-1/3)=-22/3\) (the minimum), \(f(3,-1/3)=-19/3,\) \(f(2,-3)=14,\) \(f(0,-1/3)=-10/3,\) \(f(2,0)=-7,\) \(f(0,0)=-3,\) \(f(3,0)=-6,\) \(f(3,-3)=15,\) and \(f(0,-3)=18\) (the maximum).

Example 4.29 Find the hottest and coldest points on the metal plate given as the region \(R=[0,\pi ]\times [0,\pi ],\) whose temperature is given by \(f(x,y)=\sin x+\cos 2y.\)

::: {.proof }[Solution] Since \(f\) is continuous and \(R\) is closed and bounded, we know that the absolute maximum and minimum exist. We find that \[ \nabla f=\cos x \vec{i}+(-2\sin 2y)\vec{j} \] and so the only critical point in the interior of \(R\) is \((\pi /2,\pi /2).\) This is a saddle point because the discriminant of \(f\) at \((\pi 2,\pi /2)\) is negative. The boundary of \(R\) consists of four line segments \(L_1,\) \(L_2,\) \(L_3,\) and \(L_4\) as follows.

On \(L_1\) and \(L_3,\) we have \[ f(x,0)=f(x,\pi) =\sin x+1 \quad \text{ for } 0\leq x\leq \pi \] which achieves a maximum at \(\pi /2\) and a minimum at \(0\) and \(\pi .\) Similarly, on \(L_2\) and \(L_4\) we have \[ f(0,y)=f(\pi ,y) =\cos 2y \quad \text{ for } 0\leq y\leq \pi \] which achieves its maximum at \(0\) and \(\pi\) and its minimum at \(\pi /2.\) We see that the hottest points are \((\pi /2,0)\) and \((\pi /2,\pi )\) and the coldest points are \((0,\pi /2)\) and \((\pi ,\pi /2).\) :::

4.10 Exercises

Exercise 4.10 Find the absolute extrema for the function \(f(x,y)=x y^2-2xy+3y\) in the triangular domain with vertices \((0,0)\), \((1,0)\), and \((1,1)\).

Exercise 4.11 Find and classify the relative extrema and the saddle points of \(f(x,y)=xy -2x-4y\).

Exercise 4.12 Find the critical points of the function \(f(x,y)=x^3+y^2-2xy+7x-8y+2\).

Exercise 4.13 Find and classify the relative extrema and the saddle points of \(f(x,y)=xy -2x-4y\).

Exercise 4.14 Find the absolute extrema for the function \(f(x,y)=x y^2-2xy+3y\) in the triangular domain with vertices \((0,0)\), \((1,0)\), and \((1,1)\).

Exercise 4.15 Find the critical points and classify each as a relative maximum, a relative minimum, or a saddle point.

  • \(f(x,y)=2x^2-4 x y+y^3+2\)
  • \(f(x,y)=e^{-x}\sin y\)
  • \(f(x,y)=(x-2)^2+(y-3)^4\)
  • \(f(x,y)=\left(x^2+2y^2\right)e^{1-x^2-y^2}\)
  • \(f(x,y)=x^2+y^2+\frac{32}{x y}\)
  • \(f(x,y)=(x-4)\ln (x y)\)
  • \(f(x,y)=2x^3+y^3+3x^2-18y^2+81y+5\)
  • \(f(x,y)=3x^2+12x+8y^3-12y^2+7\)

Exercise 4.16 Find the absolute maximum and minimum values for each of the following functions.

  • \(f(x,y)=e^{x^2+2x+y^2}\) on the disk \(x^2+2x+y^2\leq 0.\)
  • \(f(x,y)=x^2+x y +y^2\) on the disk \(x^2+y^2\leq 1.\)
  • \(f(x,y)=x y-2x-5y\) on the triangular region \(S\) with vertices \((0,0),\) \((7,0),\) and \((7,7).\)
  • \(f(x,y)=x^2-4 x y+y^3+4y\) on the square \(0\leq x\leq 2,\) \(0\leq y\leq 2.\)
  • \(f(x,y)=x^2+3y^2-4x+2y-3\) on the square region \(S\) with vertices \((0,0),\) \((3,0),\) \((3,-3),\) and \((0,-3).\)

Exercise 4.17 Find three positive numbers whose sum is 54 and whose product is as large as possible.

Exercise 4.18 A wire of length \(L\) is cut into three pieces that are bent to form a circle, a square, and a equilateral triangle. How should the cuts be made to minimize the sum of the total area.?

Exercise 4.19 A rectangular box with no top is to have a fixed volume. What should its dimensions be if we want to use the least amount of material in its construction?

Exercise 4.20 Let \(R\) be the triangular region in the \(x y\)-plane with vertices \((-1,-2),\) \((-1,2),\) and \((3,2).\) A plate in the shape of \(R\) is heated so that the temperature at \((x,y)\) is \[ T(x,y)=2x^2-x\text{ }y+y^2-2y+1 \] (in degrees Celsius). At what point in \(R\) or on its boundary is \(T\) maximized? What are the temperatures?

Exercise 4.21 Find the maximum and minimum values for the given function in the given closed region.

  • \(z=8x^2+4y^2+4y+5\), \(x^2+y^2\leq 1\)
  • \(z=6x^2+y^3+6y^2\), \(x^2+y^2\leq 25\)
  • \(z=8x^2-24xy+y^2\), \(x^2+y^2\leq 25\)

Exercise 4.22 Find the volume of the largest box that cane inscribed in the ellipsoid \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1. \]

Exercise 4.23 Assuming that the function \[ F= 2x^2+6y^2+45z^2-4xy+6yz+12xz-6y+14 \] has a minimum, find it. Prove that this function has no maximum value.

4.11 Lagrange Multipliers

In many applied problems, the main focus is on optimizing a function subject to constraint; for example, finding extreme values of a function of several variables where the domain is restricted to a level curve (or surface) of another function of several variables. Lagrange multipliers are a general method which can be used to solve such optimization problems.

4.12 Lagrange’s Theorem

The \(\lambda\) in \(\ref{lagthm}\) is called a Lagrange multiplier. Thus, Lagrange’s Theorem gives necessary conditions for the existence of a Lagrange multiplier.

::: {#thm- } [Lagrange]

Assume that \(f\) and \(g\) have continuous first partial derivatives and that \(f\) has an extremum at \(P_0\left(x_0,y_0\right)\) on the smooth constraint curve \(g(x,y)=c.\) If \(\nabla g\left(x_0,y_0\right)\neq \vec{0},\) there is a number \(\lambda\) such that \[\begin{equation} \label{lagmult} \nabla f\left(x_0,y_0\right)=\lambda \nabla g\left(x_0,y_0\right). \end{equation}\] :::

Proof. Denote the constraint curve \(g(x,y)=c\) by \(C\) and note that \(C\) is smooth. We represent this curve by the vector function \[ \vec{R}(t)=x(t)\vec{i}+y(t)\vec{j} \] for all \(t\) in an open interval \(I,\) including \(t_0\) corresponding to \(P_0,\) where \(x'(t)\) and \(y'(t)\) exist and are continuous. Let \(F(t)=f(x(t),y(t))\) for all \(t\) in \(I,\) and apply the chain rule to obtain \[ F'(t)=f_x(x(t),y(t))\frac{d x}{d t}+f_y(x(t),y(t))\frac{d y}{d t}=\nabla f(x(t),y(t))\cdot \vec{R}'(t). \] Because \(f(x,y)\) has an extremum at \(P_0,\) we know that \(F(t)\) has an extremum at \(t_0.\) Therefore, we have \(F'\left(t_0\right)=0\) and \[ F'\left(t_0\right)=\nabla f\left(x\left(t_0\right),y\left(t_0\right)\right)\cdot \vec{R}'\left(t_0\right)=0. \] If \(\nabla f\left(x\left(t_0\right),y\left(t_0\right)\right)=\vec{0},\) then \(\lambda =0,\) and the condition \(\nabla f=\lambda \nabla g\) is satisfied trivially. If \(\nabla f\left(x\left(t_0\right),y\left(t_0\right)\right)\neq \vec{0}\), then \(\nabla f\left(x\left(t_0\right),y\left(t_0\right)\right)\) is orthogonal to \(\vec{R}'\left(t_0\right).\) Because \(\vec{R}'\left(t_0\right)\) is tangent to the constraint curve \(C\), it follows that \(\nabla f\left(x_0,y_0\right)\) is normal to \(C.\) But \(\nabla g\left(x\left(t_0\right),y\left(t_0\right)\right)\) is also normal to \(C\) (because \(C\) is a level curve of \(g\)), and we conclude that \(\nabla f\) and \(\nabla g\) must be parallel at \(P_0.\) Thus, there is a scalar \(\lambda\) such that \(\ref{lagmult}\) holds.

Example 4.30 Find the extreme values of the function \(f(x,y)=x^2+y\) constrained to the circle \(x^2+y^2=1.\)

Solution. Using \(\ref{lagthm}\), we solve the equations \(\nabla f=\lambda \nabla g,\) \(g(x,y)=1,\) which can be written as \(f_x=\lambda g_x,\) \(f_y=\lambda g_y\), and \(g(x,y)=1\) or written as \[ 2x=2x \lambda, \qquad 1=2y \lambda, \qquad \text{and} \qquad x^2+y^2=1. \] From the first equation we have \(x=0\) or \(\lambda =1.\) If \(x=0,\) then by the third equation \(y=\pm 1.\) If \(\lambda =1,\) then \(y=1/2\) and we obtain \(x=\left.\pm \sqrt{3}\right/2.\) Therefore, \(f\) has possible extreme values at the points \[ (0,1), \qquad (0,-1), \qquad \left(\left.\sqrt{3}\right/2,1/2\right), \qquad \text{and} \qquad \left(\left.-\sqrt{3}\right/2,1/2\right). \] Evaluating \(f\) at these four points, we find that \[ f(0,1)=1, \qquad f(0,-1)=-1, \qquad \text{and} \qquad f\left(\left.\pm \sqrt{3}\right/2,1/2\right)=5/4.\] Therefore the maximum value of \(f\) constrained to the circle \(x^2+y^2=1\) is \(5/4\) and the minimum value is \(-1.\)

4.13 The Method of Lagrange Multipliers

::: {#thm- } Lagrange Multipliers

Suppose \(f\) and \(g\) satisfy the hypotheses of Lagrange’s theorem, and that \(f\) has an extremum subject to the constraint \(g(x,y)=c.\) Then to find the extreme value, proceed as follows:

  • Simultaneously solve the following three equations for \(x,y\) and \(\lambda\): \[ f_x(x,y)=\lambda g_x(x,y), \quad f_y(x,y)=\lambda g_y(x,y), \quad \text{and} \quad g(x,y)=c \]
  • Evaluate \(f\) at all points found in step (i). The extremum we seek must be among these values.

:::

%\begin{comment} We use the method of Lagrange multipliers to find the required constrained extrema. Suppose \(E\) is an extreme value of \(f\) subject to the constraint \(g(x,y)=c.\) Then the Lagrange multiplier \(\lambda\) is the rate of change of \(E\) with respect to \(c\); that is \(\lambda =d E/d c.\) Note that at the extreme value \((x,y)\) we have \(f_x=\lambda g_x\), \(f_y=\lambda g_y\), and \(g(x,y)=c.\) The coordinates of the optimal ordered pair \((x,y)\) depend on \(c\) (because different constraint levels will generally lead to different optimal combinations of \(x\) and \(y\)). Thus, \(E=E(x,y)\) where \(x\) and \(y\) are functions of \(c.\) By the chain rule for partial derivatives:
\[ \frac{d E}{d c}=f_x\frac{d x}{d c}+f_y\frac{d y}{d c}=\text{$\lambda $g}_x\frac{d x}{d c}+\text{$\lambda $g}_y\frac{d y}{d c}=\lambda \left(\frac{d g}{d c}\right)=\lambda \] %\end{comment}

Example 4.31 Maximize \(f(x,y)=x y\) subject to \(2x+2y=5.\)

Solution. Let \(g(x,y)=2x+2y,\) then we have \[ f_x=y, \qquad f_y=x, \qquad g_x=2, \qquad \text{and} \qquad g_y=2. \] We need to solve the system \[ y=2\lambda, \qquad x=2\lambda, \qquad \text{and} \qquad 2x+2y=5. \] We find \(x=y=\frac{5}{4}.\) Therefore, \(f\left(\frac{5}{4},\frac{5}{4}\right)=\frac{25}{16}\) is the constrained maximum.

Example 4.32 Minimize \(f(x,y,z)=x^2+y^2+z^2\) subject to \(x-2y+3z=4.\)

Solution. Let \(g(x,y,z)=x-2y+3z,\) then we have \[ f_x=2x, \quad f_y=2y, \quad f_z=2z, \quad g_x=1, \quad g_y=-2, \quad \text{and} \quad g_z=3. \] We need to solve the system \[ 2x=\lambda, \qquad 2y=-2\lambda, \qquad 2z=3\lambda, \qquad \text{and} \qquad x-2y+3z=4. \] We find that \(\lambda =\frac{4}{7}\) and then \(x=\frac{2}{7},\) \(y=-\frac{4}{7},\) and \(z=\frac{6}{7}.\) Therefore, \[ f\left(\frac{2}{7}, -\frac{4}{7},\frac{6}{7}\right)=\frac{8}{7} \] is the constrained minimum.

Example 4.33 A rectangular box with no top is to be constructed from 96 \(\text{ft}^2\) of material. What should be the dimensions of the box if it is to enclose maximum volume?

Solution. Let \(x,y,\) and \(z\) be the length, width, and height of the rectangular box, respectively. We want to maximize the volume: \[ V=x y z \qquad \text{subject to} \qquad S(x,y,z)=x y+2 x z+2 y z=96 \] which is obtained from the surface area of the rectangular box (with no lid). We have \[ V_x=y z, \quad V_y=x z, \quad V_z=x y, \quad S_x=y+2 z, \quad S_y=x+2 z, \quad S_z=2x+2y. \] Solve the system \[ y z=\lambda (y+2 z), \quad x z=\lambda (x+2 z), \quad x y=\lambda (2x+2 y), \quad x y+2 x z+2 y z=96. \] We obtain \(x=y=2 z,\) and then find \(x=y=4\sqrt{2},\) \(z=2\sqrt{2}.\) Therefore the maximum volume is \[ V\left(4\sqrt{2},4\sqrt{2},2\sqrt{2}\right)=64\sqrt{2} \text{ft}^3. \]

Example 4.34 A cylindrical can is to hold \(4\pi \text{in}.^3\) of orange juice. The cost per square inch of constructing the metal top and bottom is twice the cost per square inch of constructing the cardboard side. What are the dimensions of the least expensive can?

Solution. Let \(x\) and \(y\) be the radius and height of the cylinder, respectively. We want to minimize the cost \[ f(x,y)=2\left(2\pi x^2\right)+2\pi x y \quad \text{subject to the constraint} \quad g(x,y)=\pi x^2 y=4\pi. \] We have \[ f_x=8\pi x+2\pi y, \qquad f_y=2\pi x, \qquad g_x=2\pi x y, \qquad \text{and} \qquad g_y=\pi x^2. \] Solving the system \[ 8\pi x+2\pi y=2\lambda \pi x y, \qquad 2\pi x=\lambda \pi x^2, \qquad \pi x^2 y=4\pi. \]
we obtain \(y=4x,\) and then find the radius \(x=1\) in. and the height \(y=4\) in.

4.14 Optimizing a Function Subject to Two Constraints

The method of Lagrange multipliers can also be applied in situations with more than one constraint equation. Suppose we wish to locate an extremum of a function defined by \(f(x,y,z)\) subject to constraints \(g(x,y,z)=c_1\) and \(h(x,y,z)=c_2,\) where \(g\) and \(h\) are also differentiable and \(\nabla g\) and \(\nabla h\) are not parallel. By generalizing Lagrange’s theorem, it can be shown that if \(\left(x_0,y_0,z_0\right)\) is the desired extremum, then there are numbers \(\lambda\) and \(\mu\) such that \(g\left(x_0,y_0,z_0\right)=c_1, h\left(x_0,y_0,z_0\right)=c_2,\) and \[ \nabla f\left(x_0,y_0,z_0\right)=\lambda \nabla g\left(x_0,y_0,z_0\right)+\mu \nabla h\left(x_0,y_0,z_0\right). \] As in the case of one constraint, we proceed by first solving this system of equations simultaneously to find \(\lambda , \mu , x_0, y_0, \text{and} z_0\) and then evaluating \(f(x,y,z)\) at each solution and comparing to find the assumed extremum.

Example 4.35 Find the maximum value of the function \[ f(x,y,z)=x+2y+3z \] on the curve of intersection of the plane \(x-y+z=1\) and the cylinder \(x^2+y^2=1.\)

Solution. We maximize the function\(f(x,y,z)=x+2y+3z\) subject to the constraints \(g(x,y,z)=x-y+z=1\) and \(h(x,y,z)=x^2+y^2=1.\) The Lagrange condition is \(\nabla f=\lambda \nabla g+\mu \nabla h,\) so we solve the equations \[ 1=\lambda +2x \mu, \quad 2=-\lambda +2, \quad y \mu 3=\lambda, \quad x-y+z=1, \quad \text{and} \quad x^2+y^2=1.\] Putting \(\lambda =3,\) we get \(2x \mu =-2,\) so \(x=-1/\mu .\) Similarly, we have \(y=5/(2\mu ).\) Substitution yields \[\frac{1}{\mu ^2}+\frac{25}{4\mu^2}=1\] and so \(u^2=\frac{29}{4}.\) Then \[ \mu =\left.\pm \sqrt{29}\right/2, \qquad x=\mp 2\left/\sqrt{29}\right., \qquad \text{and} \qquad y=\pm 5\left/\sqrt{29}\right.\] and so we have \(z=1-x+y\) \(=1\pm 7\left/\sqrt{29}.\right.\) The corresponding values of \(f\) are \[ \mp \frac{2}{\sqrt{29}}+2\left(\pm \frac{5}{\sqrt{29}}\right)+3\left(1\pm \frac{7}{\sqrt{29}}\right)=3\pm \sqrt{29}.\] Therefore the maximum of \(f\) on the given curve is \(3+\sqrt{29}.\)

Example 4.36 Find the maximum of \(f(x,y,z)=x y z\) subject to \(x^2+y^2=3\) and \(y=2 z.\)

Solution. We need to solve the system \(\nabla f=\lambda \nabla g+\mu \nabla h\) where \(g(x,y,z)=x^2+y^2\) and \(h(x,y,z)=y-2z.\) Therefore we need to solve the system \[ \begin{array}{lll} \left\{ \begin{array}{l} f_x-\lambda g_x-\mu h_x=0 \\ f_y-\lambda g_y-\mu h_y=0 \\ f_z-\lambda g_z-\mu h_z=0 \\ x^2+y^2-3=0 \\ y-2z=0 \end{array} \right. & \text{ which is written as } & \left\{ \begin{array}{l} y z-2\lambda x=0 \\ x z-2\lambda y+\mu =0 \\ x y+2\mu =0 \\ y-2z=0 \\ x^2+y^2-3=0. \end{array} \right. \end{array}\] The solutions are \(\left(0,0,\pm\sqrt{3}\right)\), and \[ \left(1,\sqrt{2},\frac{\sqrt{2}}{2}\right), \quad \left(1,-\sqrt{2},-\frac{\sqrt{2}}{2}\right), \quad \left(-1,\sqrt{2},\frac{\sqrt{2}}{2}\right), \quad \left(-1,-\sqrt{2},-\frac{\sqrt{2}}{2}\right). \] The maximum is \(f\left(1,\sqrt{2},\frac{\sqrt{2}}{2}\right)=f\left(1,-\sqrt{2},-\frac{\sqrt{2}}{2}\right)=1.\)

Example 4.37 Find the minimum of \[ f(x,y,z)=x^2+y^2+z^2 \] subject to \(x+y=4\) and \(y+z=6.\)

Solution. We want to minimize \(x^2+y^2+z^2\) subject to the side conditions \(x+y-4=0\) and \(y+z-6=0.\) We form \[ L(x,y,z,\lambda ,\mu )=x^2+y^2+z^2-\lambda (x+y-4)-\mu (y+z-6).\] The conditions are \(\frac{\partial L}{\partial x}=2x-\lambda =0\) and \[ \begin{array}{lll} \frac{\partial L}{\partial y}=2y-\lambda -\mu =0 & \qquad \qquad & \frac{\partial L}{\partial z}=2z-\mu =0 \\ \frac{\partial L}{\partial \lambda }=x+y-4=0 & & \frac{\partial L}{\partial \mu }= y+z-6=0. \end{array} \] The first and third conditions give \(\lambda =2x\) and \(\mu =2z,\) so the second condition becomes \(2y-2x-2z=0.\) We then have \[ \left\{\begin{array}{l} x+ y+z = 0 \\ x+ y = 4 \\ y+z = 6 \end{array} \right. \] The solution to this system is \(P=(2/3,10/3,8/3).\) Therefore the minimum is \(f(P)=56/3.\)

Example 4.38 Use Lagrange multipliers to find the point on the line of intersection of the planes \(x-y=2\) and \(x-2z=4\) that is closest to the origin.

Solution. We want to minimize \(x^2+y^2+z^2\) subject to the side conditions \(x-y-2=0\) and \(x-2z-4=0.\) We form \[ L(x,y,z,\lambda ,\mu )=x^2+y^2+z^2-\lambda (x-y-2)-\mu (x-2z-4). \] The conditions are \(\frac{\partial L}{\partial x}=2x-\lambda -\mu =0\) and \[ \begin{array}{lll} \frac{\partial L}{\partial y}=2y+\lambda =0 & \qquad \qquad & \frac{\partial L}{\partial z}=2z+2\mu =0 \\ \frac{\partial L}{\partial \lambda }=-x+y+2=0 & & \frac{\partial L}{\partial \mu }=-x+2z+4=0. \end{array} \] The second and third conditions give \(\lambda =-2y\) and \(\mu =-z,\) so the first condition becomes \(2x+2y+z=0.\) We then have \(2x+2y+z =0\), \(-x+y=-2\), \(-x+2z=-4.\) The last two equations may be written as \(y=x-2\) and \(z=(x-4)/2.\) Substitution of these values into the first equation gives \(x=4/3.\) Consequently, \(y=-2/3\) and \(z=-4/3.\) The desired point is therefore, \((4/3,-2/3,-4/3).\)

Example 4.39 Find the maximum and minimum values of \[ f(x,y,z)=5x-y-6z \] on the surface \(2x^2+4y^2+6z^2=200.\)

Solution. We set \(g(x,y,z)=2x^2+4y^2+6z^2\) and we use the Lagrange multiplier \(\lambda\) and solve the system
\[ \begin{array}{ccc} \left\{ \begin{array}{l} f_x = \lambda g_x \\ f_y = \lambda g_y \\ f_z = \lambda g_z \\ g = 200 \end{array} \right. & \text{which is written as } & \left\{ \begin{array}{l} 5 = 4x \lambda \\ -1 = 8y \lambda \\ -6 = 12 z \lambda \\ 2x^2+4y^2+6z^2 = 200 \end{array} \right. \end{array} \] to find
\[ P=\left(-10 \sqrt{\frac{2}{3}},\sqrt{\frac{2}{3}},4 \sqrt{\frac{2}{3}} \right) \qquad \text{and} \qquad Q=\left(10 \sqrt{\frac{2}{3}},-\sqrt{\frac{2}{3}},-4 \sqrt{\frac{2}{3}}\right) \] Therefore,
\(f(P)=-25\sqrt{6}\) is the minimum and \(f(Q)=25\sqrt{6}\)
is the maximum value.

Example 4.40 Maximize \[ f(x,y)=\ln \left(x y^2\right) \] subject to the constraint \(2x^2+3y^2=8\) for \(x>0.\)

Solution. Let \(g(x,y)=2x^2+3y^2\) and we will use a Lagrange multiplier, say \(\lambda\). We setup the system of equations
\[ \begin{array}{ccc} \left\{ \begin{array}{l} \frac{ \partial f}{\partial x} = \lambda \frac{\partial g}{\partial x} \\ \frac{ \partial f}{\partial y} = \lambda \frac{\partial g}{\partial y} \\ g(x) = 8 \end{array} \right. & \text{which is written as } & \left\{ \begin{array}{l} \frac{y^2}{x y^2} = 4\lambda x \\ \frac{2x y}{x y^2} = 6\lambda y \\ 2x^2+3y^2 = 8 \end{array} \right. \end{array} \] Now to solve this we will try to eliminate \(\lambda\) from the first two equations. So we solve both of them for \(\lambda\) we have,
\[ \lambda =\frac{1}{4 x^2} \qquad \text{and} \qquad \lambda =\frac{1}{3 x^2 y}, \] (since \(x>0\)) respectively. Thus, these two expressions must equal yielding \[ \frac{1}{4 x^2}=\frac{1}{3 x^2 y} \] we have \(4x^2=3x^2y\) which we write as \(x^2(4-3y)=0\) and so \(y=4/3.\) So that \[ x^2=\frac{8-3(4/3)^2}{2}=\frac{4}{3}.\] However, since \(x>0\) we only use \(x=\frac{2}{\sqrt{3}}\) with \(y=4/3;\) and therefore the maximum value of \(f(x,y)=\ln \left(x y^2\right)\) subject to the constraint \(g(x,y)=8\) is \[ f\left(\frac{2}{\sqrt{3}},\pm \frac{4}{3}\right)=\ln \left(\frac{2}{\sqrt{3}}\left(\frac{4}{3}\right)^2\right)=\frac{5}{2}\ln \left(\frac{4}{3}\right). \]

4.15 Exercises

Exercise 4.24 Use Lagrange multipliers to find the point on the line of intersection of the planes \(x-y=2\) and \(x-2z=4\) that is closest to the origin.

Exercise 4.25 Use the method of Lagrange multipliers to maximize or minimize each of the functions.

  • maximize \(f(x,y)=x y\) subject to \(2x+2y=5\)
  • minimize \(f(x,y)=x y z\) subject to \(3x+2y+z=6\)
  • maximize \(f(x,y)=e^{x y}\) subject to \(x^2+y^2=3\)
  • maximize \(f(x,y)=\cos x+\cos y\) subject to \(y=x+\frac{\pi }{4}\)
  • minimize \(f(x,y)=x^2-x y+2y^2\) subject to \(2x+y=22\)
  • minimize \(f(x,y)=x^2-y^2\) subject to \(x^2+y^2=4\)
  • minimize \(f(x,y,z)=2x^2+3y^2+4z^2\) subject to \(x+y+z=4\) and \(x-2y+5z=3\)
  • minimize \(f(x,y,z)=x^2+y^2+z^2\) subject to \(x+y=4\) and \(y+z=6\)
  • minimize \(f(x,y,z)=x y z\) subject to \(x^2+y^2=3\) and \(y=2z\)
  • maximize \(f(x,y,z)=x y+x z\) subject to \(2x+3z=5\) and \(x y=4\)

Exercise 4.26 Find the point on the plane \(2x-3y+6z=5\) that is closest to the origin.

Exercise 4.27 Find the point on the plane \(3x-2y+z=5\) that is closet to \((4,-8,5)\).

Exercise 4.28 A rectangular box has a square base and one top. Find the dimensions for minimum surface area the volume is to be 108 in\(^3\).

Exercise 4.29 An open rectangular box has ends costing \(\$6/\text{ft}^2\), sides costing \(\$4/\text{ft}^2\), and a bottom costing \(\$10/\text{ft}^2\). Find the dimensions for the minimum cost if the volume of the box is 120 ft\(^3\).

Exercise 4.30 Find the maximum possible volume of a right circular cone inscribed in a sphere of radius \(a\).

Exercise 4.31 The sides of a closed cylindrical container cost twice as much per square food as the ends. Find the ratio of the radius to the altitude of the cylinder for the cheapest such container having fixed volume.

Exercise 4.32 A pentagon consists of a rectangle surmounted by an isosceles triangle. Find the dimensions of the pentagon having the maximum area if the perimeter is to be \(P\).

Exercise 4.33 Find the point on the curve of intersection of \(x^2+z^2=4\) and \(x-y=8\) that is farthest form the origin.

Exercise 4.34 Find the point on the line of intersection of the planes \(x-y=4\) and \(y+3z=6\) that is closet to \((-1,3,2)\).