Second derivative test
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In calculus, a branch of mathematics, the second derivative test determines whether a given stationary point of a function is a maximum or a minimum.
The test states: If the function f is twice differentiable in a neighbourhood of a stationary point x, then:
- If f''(x) < 0 then f has a maximum at x.
- If f''(x) > 0 then f has a minimum at x.
Note that if f''(x) = 0 the second derivative test says nothing about the point x.
[edit] Multivariable case
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 stationary point. In particular, assuming that all second order partial derivatives of f are continuous on a neighbourhood of a stationary 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.
[edit] See also
- First derivative test
- Higher order derivative test
- Differentiability
- Extreme value
- Inflection point
- Convex function
- Concave function
