8. [IMPLICIT METHOD].

(a) Suppose that

f, g : R3 - R are differentiable at a point

(a,

b, c), and f

(a,

b, c)

is an extremum of f subject to the constraint g(x, y, z) = k, where k is a constant. Prove that and of og of og oy

(a,

b, c) oz

(a,

b, c) - oz

(a,

b, c) oy

(a,

b, c) = O.

(b) Use part

(a) to find all extrema of f(x, y, z) = 4xy + 2xz + 2yz subject to the constraint xyz = 16.

[!]. This exercise is used in Section e14.4.

(a) Let p > 1. Find all extrema of f(x) = E~=l x~ subject to the constraint E~=l IXklP = 1.

(b) Prove that for all Xl, .•• ,Xn E R, n E N, and 1 ~ p ~ 2