If f is a differentiable function and f'(c) = 0, then at x = c, f has

Core Concept

If f'(c) = 0, then c is a critical point of f. At a critical point, f can have either a relative maximum, a relative minimum, or neither (as in the case of f(x) = x^3 at x = 0, which has a horizontal tangent but no extremum). Therefore, if f'(c) = 0, f could have either a relative maximum or a relative minimum at x = c, but it's not guaranteed to have one without further information.