Exercises for Section 14.4

Question 14.107

Let \(f(x, y)=x^2+3y^2\). Find the maximum and minimum values of \(f\) subject to the given constraint.

  • (a) \( x^2+y^2=1\)
  • (b) \( x^2+y^2 \leq 1\)

Question 14.108

Consider all rectangles with fixed perimeter \(p\). Use Lagrange multipliers to show that the rectangle with maximal area is a square.

In Exercises 3 to 7, find the extrema of f subject to the stated constraints.

Question 14.109

\(f(x,y,z) = x-y +z\), subject to \(x^2 + y^2 + z^2 = 2\)

Question 14.110

\(f(x,y) = x- y\), subject to \(x^2 - y^2 = 2\)

Question 14.111

\(f(x,y) = x\), subject to \(x^2 + 2y^2 = 3\)

Question 14.112

\(f(x, y, z) = x+ y + z\), subject to \(x^2 - y^2 = 1, 2x + z =1\)

Question 14.113

\(f(x,y) = 3x + 2y\), subject to \(2x^2 + 3y^2 =3\)

Find the relative extrema of \(f{\mid} S\) in Exercises 8 to 11.

Question 14.114

\(f{:}\,\, {\mathbb R}^2 \rightarrow {\mathbb R}, (x,y) \mapsto x^2 + y^2, S = \{(x,2) \mid x \in {\mathbb R} \}\)

Question 14.115

\(f{:}\,\, {\mathbb R}^2 \rightarrow {\mathbb R}, (x,y) \mapsto x^2 + y^2, S = \{(x,y) \mid y \geq 2 \}\)

Question 14.116

\(f{:}\,\, {\mathbb R}^2 \rightarrow {\mathbb R}, (x,y) \mapsto x^2 - y^2, S = \{(x,\cos x) \mid x \in {\mathbb R} \}\)

Question 14.117

\(f{:}\,\, {\mathbb R}^3 \rightarrow {\mathbb R}, (x,y,z) \mapsto x^2 + y^2 + z^2, S = \{(x,y,z) \mid z \geq 2 + x^2 + y^2 \}\)

Question 14.118

Use the method of Lagrange multipliers to find the absolute maximum and minimum values of \(f(x,y) = x^2 + y^2 -x - y +1\) on the unit disc (see Example 10 of Section 14.3).

Question 14.119

Consider the function \(f(x,y) = x^2 + xy + y^2\) defined on the unit disc, namely, \(D = \{(x,y) \mid x^2 + y^2 \leq 1\}\). Use the method of Lagrange multipliers to locate the maximum and minimum points for \(f\) on the unit circle. Use this to determine the absolute maximum and minimum values for \(f\) on \(D\).

202

Question 14.120

Find the absolute maximum and minimum values of \(f(x, y, z)=2x+y\), subject to the constraint \(x+y+z=1\).

Question 14.121

Find the extrema of \(f(x, y)=4x+2y\), subject to the constraint \(2x^2+3y^2=21\).

Question 14.122

Use Lagrange multipliers to find the distance from the point (2, 0, \(-1\)) to the plane \(3x-2y+8z+1=0\). Compare your answer to Example 12 in Section 1.3.

Question 14.123

Find the maximum and minimum values attained by \(f(x, y, z)=xyz\) on the unit ball \(x^2+y^2+z^2 \leq 1\).

Question 14.124

Let \(S\) be the sphere of radius 1 centered at (1, 2, 3). Find the distance from \(S\) to the plane \(x+y+z=0\). (HINT: Use Lagrange multipliers to find the distance from the plane to the center of the sphere.)

Question 14.125

  • (a) Find three numbers whose product is 27 and whose sum is minimal.
  • (b) Find three numbers whose sum is 27 and whose product is maximal.

Question 14.126

A rectangular box with no top is to have a surface area of 16 m\(^2\). Find the dimensions that maximize its volume.

Question 14.127

Design a cylindrical can (with a lid) to contain 1 liter (\({=}\,\)1000 cm\(^{3}\)) of water, using the minimum amount of metal.

Question 14.128

Show that solutions of equations (4) and (5) are in one-to-one correspondence with the critical points of \begin{eqnarray*} && h(x_1,\ldots, x_n, \lambda_1, \ldots, \lambda_k) \\ &&\quad = f(x_1, \ldots,x_n) -\lambda_1 [g_1(x_1,\ldots,x_n)\,{-}\,c_1]\\ &&\quad - \cdots - \lambda_k [g_k(x_1,\ldots,x_n)-c_k]. \end{eqnarray*}

Question 14.129

Find the absolute maximum and minimum for the function \(f(x,y,z)=x+y-z\) on the ball \(B=\{(x,y,z)\mid x^2+y^2+z^2\leq 1\}\).

Question 14.130

Repeat Exercise 23 for \(f(x,y,z)=x+yz\).

Question 14.131

A rectangular mirror with area \(A\) square feet is to have trim along the edges. If the trim along the horizontal edges costs \(p\) cents per foot and that for the vertical edges costs \(q\) cents per foot, find the dimensions that will minimize the total cost.

Question 14.132

An irrigation canal in Arizona has concrete sides and bottom with trapezoidal cross section of area \(A=y(x + y \tan \theta)\) and wetted perimeter \(P=x + 2y/ \cos \theta\), where \(x=\) bottom width, \(y=\) water depth, and \(\theta =\) side inclination, measured from vertical. The best design for a fixed inclination \(\theta\) is found by solving \(P=\) minimum subject to the condition \(A = \hbox{constant}\). Show that \(y^2 = (A \cos \theta)/(2 - \sin \theta)\).

Question 14.133

Apply the second-derivative test to study the nature of the extrema in Exercises 3 and 7.

Question 14.134

A light ray travels from point A to point B crossing a boundary between two media (see Figure 14.35). In the first medium its speed is \(v_1\), and in the second it is \(v_2\). Show that the trip is made in minimum time when Snell’s law holds: \[ \frac{\sin \theta_1}{\sin \theta_2} = \frac{v_1}{v_2}. \]

Question 14.135

A parcel delivery service requires that the dimensions of a rectangular box be such that the length plus twice the width plus twice the height be no more than 108 inches \((l + 2w +\) \(2h \leq 108)\). What is the volume of the largest-volume box the company will deliver?

Figure 14.35: Snell’s law of refraction.

Question 14.136

Let P be a point on a surface \(S\) in \({\mathbb R}^3\) defined by the equation \(f(x,y,z)=1\), where \(f\) is of class \(C^1\). Suppose that P is a point where the distance from the origin to \(S\) is maximized. Show that the vector emanating from the origin and ending at P is perpendicular to \(S\).

Question 14.137

Let \(A\) be a nonzero symmetric \(3 \times 3\) matrix. Thus, its entries satisfy \(a_{\it ij} = a_{ji}\). Consider the function \(f({\bf x}) = \frac{1}{2} (A{\bf x})\, {\cdot}\, {\bf x}\).

  • (a) What is \({\nabla} f\)?
  • (b) Consider the restriction of \(f\) to the unit sphere \(S = \{(x,y,z) \mid x^2 + y^2 + z^2 = 1\}\) in \({\mathbb R}^3\). By Theorem 7 we know that \(f\) must have a maximum and a minimum on \(S\). Show that there must be an \({\bf x}\in S\) and a \(\lambda \neq 0\) such that \(A{\bf x} = \lambda{\bf x}.\) (The vector \({\bf x}\) is called an eigenvector, while the scalar \(\lambda\) is called an eigenvalue.)
  • (c) What are the maxima and minima for \(f\) on \(B=\{(x,y,z)\mid x^2+y^2+z^2\leq 1\}\)?

203

Question 14.138

Suppose that \(A\) in the function \(f\) defined in Exercise 31 is not necessarily symmetric.

  • (a) What is \({\nabla} f\)?
  • (b) Can we conclude the existence of an eigenvector and eigenvalues as in Exercise 31?

Question 14.139

  • (a) Find the critical points of \(x+y^2\), subject to the constraint \(2x^2 + y^2 = 1\).
  • (b) Use the bordered Hessian to classify the critical points.

Question 14.140

Answer the question posed in the last line of Example 9.

Question 14.141

Try to find the extrema of \(xy + yz\) among points satisfying \(xz =1\).

Question 14.142

A company’s production function is \(Q(x,y) = xy\). The cost of production is \(C(x,y) = 2x + 3y\). If this company can spend \(C(x,y) =10\), what is the maximum quantity that can be produced?

Question 14.143

Find the point on the curve \((\cos t, \sin t, \sin (t/2))\) that is farthest from the origin.

Question 14.144

A firm uses wool and cotton fiber to produce cloth. The amount of cloth produced is given by \(Q(x,y) = xy - x - y +1\), where \(x\) is the number of pounds of wool, \(y\) the number of pounds of cotton, \(x>1\), and \(y>1\). If wool costs \(p\) dollars per pound, cotton costs \(q\) dollars per pound, and the firm can spend \(B\) dollars on material, what should the ratio of cotton and wool be to produce the most cloth?

Question 14.145

Carry out the analysis of Example 10 for the production function \(Q(K,L)= AK^{\alpha}L^{1-\alpha} \), where \(A\) and \(\alpha\) are positive constants and \(0 < \alpha < 1\). This is called a Cobb–Douglas production function and is sometimes used as a simple model for the national economy. \(Q\) is then the aggregate output of the economy for a given input of capital and labor.