502 CHAPTER 23. OPTIMIZATION

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

(≡ x21 × x2

2 × x23 ×·· ·× x2

n)

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

show that the maximum is achieved when x2p = y2q and equals r2. Now concludethat if x,y > 0, then

xy ≤ xp

p+

yq

q

and there are values of x and y where this inequality is an equation.

33. The area of the ellipse x2/a2 + y2/b2 ≤ 1 is πab which is given to equal π . The

length of the ellipse is∫ 2π

0

√a2 sin2 (t)+b2 cos2 (t)dt. Find a,b such that the ellipse

having this volume is as short as possible.