344 CHAPTER 18. OPTIMIZATION

23. Minimize xyz subject to the constraints x2 + y2 + z2 = r2 and x− y = 0.

24. Let n be a positive integer. Find n numbers whose sum is 8n and the sum of thesquares is as small as possible.

25. Find the point on the level surface 2x2 + xy+ z2 = 16 which is closest to (0,0,0).

26. Find the point on x2 + y2 + z2 = 1 closest to the plane x+ y+ z = 10.

27. Find the point on x2

4 + y2

9 + z2 = 1 closest to the plane x+ y+ z = 10.

28. Let x1, · · · ,x5 be 5 positive numbers. Maximize their product subject to the constraintthat

x1 +2x2 +3x3 +4x4 +5x5 = 300.

29. Let f (x1, · · · ,xn) = xn1xn−1

2 · · ·x1n. Then f achieves a maximum on the set S≡{

x ∈ Rn :n

∑i=1

ixi = 1,each xi ≥ 0

}

If x ∈ S is the point where this maximum is achieved, find x1/xn.

30. ∗ Let (x,y) be a point on the ellipse, x2/a2 +y2/b2 = 1 which is in the first quadrant.Extend the tangent line through (x,y) till it intersects the x and y axes and let A(x,y)denote the area of the triangle formed by this line and the two coordinate axes. Findthe minimum value of the area of this triangle as a function of a and b.

31. Maximize ∏ni=1 x2

i(≡ x2

1× x22× x2

3×·· ·× x2n)

subject to the constraint, ∑ni=1 x2

i = r2. Show that the maximum is(r2/n

)n. Nowshow from this that (

n

∏i=1

x2i

)1/n

≤ 1n

n

∑i=1

x2i

and finally, conclude that if each number xi ≥ 0, then(n

∏i=1

xi

)1/n

≤ 1n

n

∑i=1

xi

and there exist values of the xi for which equality holds. This says the “geometricmean” is always smaller than the arithmetic mean.

32. Maximize x2y2 subject to the constraint

x2p

p+

y2q

q= r2

where p,q are real numbers larger than 1 which have the property that

1p+

1q= 1