Let f(x) be a real-valued non-constant function and x_0 be a number in the domain of f(x). Suppose f(x) is also twice differentiable at x_0 with continuous second derivative f''(x) at x_0. 1.If f'(x_0) = 0 and f''(x_0)>0, then f(x) has a local minimum at x_0. 2.If f'(x_0) = 0 and f''(x_0)<0, then f(x) has a local maximum at x_0. 3.If f''(x_0) = 0, then the test is inconclusive.

derivative | open interval | continuous function | differentiable function | local minimum | local maximum | critical number | inflection point

first derivative test | Fermat's theorem on stationary points

Pierre de Fermat