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In calculus, the **second derivative test** is a criterion for determining whether a given critical point of a real function of one variable is a local maximum or a local minimum using the value of the second derivative at the point.

The test states: if the function *f* is twice differentiable at a critical point *x* (i.e. *f'(x) = 0*), then:

- If then has a local maximum at .
- If then has a local minimum at .
- If , the test is inconclusive.

In the latter case, Taylor's Theorem may be used to determine the behavior of *f* near *x* using higher derivatives.

Main article: Second partial derivative test

For a function of more than one variable, the second derivative test generalizes to a test based on the eigenvalues of the function's Hessian matrix at the critical point. In particular, assuming that all second order partial derivatives of *f* are continuous on a neighbourhood of a critical point *x*, then if the eigenvalues of the Hessian at *x* are all positive, then *x* is a local minimum. If the eigenvalues are all negative, then *x* is a local maximum, and if some are positive and some negative, then the point is a saddle point. If the Hessian matrix is singular, then the second derivative test is inconclusive.

Suppose we have (the proof for is analogous). By assumption, . Then

Thus, for *h* sufficiently small we get

which means that if *h* < 0 (intuitively, f is decreasing as it approaches *x* from the left), and that if *h* > 0 (intuitively, f is increasing as we go right from *x*). Now, by the first derivative test, has a local minimum at .

A related but distinct use of second derivatives is to determine whether a function is concave up or concave down at a point. It does not, however, provide information about inflection points. Specifically, a twice-differentiable function *f* is concave up if and concave down if . Note that if , then has zero second derivative, yet is not an inflection point, so the second derivative alone does not give enough information to determine if a given point is an inflection point.

- Bordered Hessian
- First derivative test
- Optimization (mathematics)
- Fermat's theorem
- Higher-order derivative test
- Differentiability
- Extreme value
- Convex function