let s(1) be the set of all critical points of f1(x, y) = g1(x).g2(y)

let-s-1-be-the-set-of-all-critical-points-of-f1-x-y-g1-x-g2-y

let s(1) be the set of all critical points of f1(x, y) = g1(x).g2(y) and s(2) be the set of critical points of f2(g1(x), g2(y)) Which of the following is the right relation between s(1) and s(2), given that minimum number of elements in s(1) is 2.

A. s(1) = s(2)

B. s(1) ≠ s(2)

C. s(1) ∩ s(2) ≠ 0

D. depends on the functions

This question was addressed to me in quiz.

The query is from Maxima and Minima of Two Variables in portion Maxima and Minima of Engineering Mathematics

The correct choice is B. s(1) ≠ s(2)

To explain I would say: Differentiating f1(g1(x), g2(y))  with respect to x and y separately we get

dx = f1x g1x (x)

dy = f1y g1y (y)

This implies

g1x = 0

g1y = 0

Which are also the set of critical points of f1(x, y)

Thus we have the relation as s(1) ∩ s(2) ≠ 0.