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
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)
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.