18.2. EXERCISES 331

Therefore, yx2− 64 = 0 and xy2− 64 = 0 so xy2 = yx2. For sure the answer excludes thecase where any of the variables equals zero. Therefore, x = y and so x = 4 = y. Then z = 2from the requirement that xyz = 32. How do you know this gives the least surface area?Why is this not the largest surface area?

18.2 Exercises1. Find the points where possible local minima or local maxima occur in the following

functions.

(a) x2−2x+5+ y2−4y

(b) −xy+ y2− y+ x

(c) 3x2−4xy+2y2−2y+2x

(d) cos(x)+ sin(2y)

(e) x4−4x3y+6x2y2−4xy3 + y4 + x2−2x

(f) y2x2−2xy2 + y2

2. Find the volume of the largest box which can be inscribed in a sphere of radius a.

3. Find in terms of a,b,c the volume of the largest box which can be inscribed in theellipsoid x2

a2 +y2

b2 +z2

c2 = 1.

4. Find three numbers which add to 36 whose product is as large as possible.

5. Find three numbers x,y,z such that x2+y2+z2 = 1 and x+y+z is as large as possible.

6. Find three numbers x,y,z such that x2 + y2 + z2 = 4 and xyz is as large as possible.

7. A feeding trough in the form of a trapezoid with equal base angles is made from along rectangular piece of metal of width 24 inches by bending up equal strips alongboth sides. Find the base angles and the width of these strips which will maximizethe volume of the feeding trough.

8. An open box (no top) is to contain 40 cubic feet. The material for the bottom coststwice as much as the material for the sides. Find the dimensions of the box which ischeapest.

9. The function f (x,y) = 2x2+y2 is defined on the disk x2+y2 ≤ 1. Find its maximumvalue.

10. Find the point on the surface z = x2 + y+1 which is closest to (0,0,0).

11. Let L1 = (t,2t,3− t) and L2 = (2s,s+2,4− s) be two lines. Find a pair of points,one on the first line and the other on the second such that these two points are closertogether than any other pair of points on the two lines.