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Second Derivative Test

Theorem

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.

Details

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

first derivative test | Fermat's theorem on stationary points

Associated person

Pierre de Fermat

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