maximum and minimum values

Question

T/F (with justification) If f(x) is a differentiable function on (a, b) and f(x) has a local maximum or minimum value at x=c in (a, b) then f'(c) =0

Answer 1

HI Emery

All extrema occur at critical points, so at first glance this looks true

However, critical points occur when the first derivative is equal to zero OR the first derivative is undefined. We can't say which, so this statement doesn't have to be true. There could be a cusp or corner at x=c.

Hope that helps!