Give an example of a function for which f'(x)= 0 does not yield minimum or maximum.

Answer 1

Any constant function, e.g., f(x) = 287

Answer 2

f(x) = x³

f'(x) = 3x² = 0 when x = 0. So, f has a critical point when x = 0.

f'(x) > 0 for x < 0 and for x > 0

Since f'(x) does not change sign at x = 0, there is neither a relative max nor a relative min at x = 0.