APPLICATION OF PARTIAL DIFFERENTIATION TO MAXIMUM AND MINIMUM PROBLEMS

8. APPLICATION OF PARTIAL DIFFERENTIATION TO MAXIMUM AND MINIMUM PROBLEMS

You will recall that derivatives give slopes as well as rates and that you find max- imum and minimum points of y = f (x) by setting dy/dx = 0. Often in applied problems we want to find maxima or minima of functions of more than one variable. Think of z = f (x, y) which represents a surface. If there is a maximum point on it (like the top of a hill), then the curves for x = const. and y = const. which pass through the maximum point also have maxima at the same point. That is, ∂z/∂x

212 Partial Differentiation Chapter 4

and ∂z/∂y are zero at the maximum point. Recall that dy/dx = 0 was a necessary condition for a maximum point of y = f (x), but not sufficient; the point might have been a minimum or perhaps a point of inflection with a horizontal tangent. Something similar can happen for z = f (x, y). The point where ∂z/∂x = 0 and ∂z/∂y = 0 may be a maximum point, a minimum point, or neither. (An interesting example of neither is a “saddle point”—a curve from front to back on a saddle has

a minimum; one from side to side has a maximum. See Figure 8.1.) In finding max- ima of y = f (x), it is sometimes possible to tell from the geometry or physics that you have a maximum.

If necessary you can find d 2 y/dx 2 ; if it is negative,

then you know you have a maximum point. There is

a similar (rather complicated) second derivative test for functions of two variables (see Problems 1 to 7), but we use it only if we have to; usually we can tell from the problem whether we have a maximum, a minimum, or neither. Let us consider some examples of maximum or minimum problems.

Figure 8.1

Example.

A pup tent (Figure 8.2) of given volume V , with ends but no floor, is to be made using the least possible material. Find the proportions.

Using the letters indicated in the figure, we find the volume V and the area A.

Figure 8.2 Since V is given, only two of the three variables w, l, and θ are independent, and

cos θ

we must eliminate one of them from A before we try to minimize A. Solving the V equation for l and substituting into A, we get

w We now have A as a function of two independent variables w and θ. To minimize

cos θ w 2

tan θ

A we find ∂A/∂w and ∂A/∂θ and set them equal to zero.

w Solving each of these equations for w 3 and setting the results equal, we get

sin 2 θ You should convince yourself that neither sin θ = 0 nor cos θ = 0 is possible (the

2 or θ = 45 ◦ . Then tan θ = 1, V=w 2

2. Then the height of the tent (at the peak) is w tan θ = w = 1/ 2.

l, and from the ∂A/∂w equation we have 2w = l √

Section 8 Application of Partial Differentiation to Maximum and Minimum Problems 213

PROBLEMS, SECTION 8

1. Use the Taylor series about x = a to verify the familiar “second derivative test” for a maximum or minimum point. That is, show that if f ′ (a) = 0, then f ′′ (a) > 0 implies a minimum point at x = a and f ′′ (a) < 0 implies a maximum point at x = a. Hint: For a minimum point, say, you must show that f (x) > f (a) for all x near enough to a.

2. Using the two-variable Taylor series [say (2.7)] prove the following “second derivative tests” for maximum or minimum points of functions of two variables. If f x =f y =0 at (a, b), then

(a, b) is a minimum point if at (a, b), f xx > 0, f yy > 0, and f xx f yy >f 2 xy ; (a, b) is a maximum point if at (a, b), f xx < 0, f yy < 0, and f xx f yy >f 2 xy ;

(a, b) is neither a maximum nor a minimum point if f xx f yy <f xy 2 . (Note that this includes f xx f yy < 0, that is, f xx and f yy of opposite sign.)

Hint: Let f xx = A, f xy = B, f yy = C; then the second derivative terms in the Taylor series are Ah 2 + 2Bhk + Ck 2 ; this can be written A(h + Bk/A) 2 + (C − B 2 /A)k 2 . Find out when this expression is positive for all small h, k [that is, all (x, y) near (a, b)]; also find out when it is negative for all small h, k, and when it has both positive and negative values for small h, k.

Use the facts stated in Problem 2 to find the maximum and minimum points of the functions in Problems 3 to 6.

3. x 2 +y 2 + 2x − 4y + 10 4. x 2 −y 2 + 2x − 4y + 10

5. 4+x+y−x 1 2 − xy − y 2 6. x 3 3

2 −y − 2xy + 2 7. 2 Given z = (y − x 2 )(y − 2x ), show that z has neither a maximum nor a minimum

at (0, 0), although z has a minimum on every straight line through (0, 0).

8. A roof gutter is to be made from a long strip of sheet metal, 24 cm wide, by bending up equal amounts at each side through equal angles. Find the angle and the dimensions that will make the carrying capacity of the gutter as large as possible.

9. An aquarium with rectangular sides and bottom (and no top) is to hold 5 gal. Find its proportions so that it will use the least amount of material.

10. Repeat Problem 9 if the bottom is to be three times as thick as the sides. 11. Find the most economical proportions for a tent as in the figure, with no floor.

12. Find the shortest distance from the origin to the surface z = xy + 5.

214 Partial Differentiation Chapter 4

13. Given particles of masses m, 2m, and 3m at the points (0, 1), (1, 0), and (2, 3), find the point P about which their total moment of inertia will be least. (Recall that to find the moment of inertia of m about P , you multiply m by the square of its distance from P .)

14. Repeat Problem 13 for masses m 1 ,m 2 ,m 3 at (x 1 ,y 1 ), (x 2 ,y 2 ), (x 3 ,y 3 ). Show that the point you find is the center of mass.

15. Find the point on the line through (1, 0, 0) and (0, 1, 0) that is closest to the line x = y = z. Also find the point on the line x = y = z that is closest to the line through (1, 0, 0) and (0, 1, 0).

16. To find the best straight line fit to a set of data points (x n ,y n ) in the “least squares” sense means the following: Assume that the equation of the line is y = mx + b and verify that the vertical deviation of the line from the point (x n ,y n ) is y n −(mx n +b). Write S = sum of the squares of the deviations, substitute the given values of x n ,y n to give S as a function of m and b, and then find m and b to minimize S.

Carry through this routine for the set of points: (−1, −2), (0, 0), (1, 3). Check your results by computer, and also computer plot (on the same axes) the given points and the approximating line.

17. Repeat Problem 16 for each of the following sets of data points. (a)