Sources
Help build the largest human-edited directory on the web.
Submit a site - Become an editor
Wikipedia. Licensed under the GNU Free Documentation License.
Are you an expert in this subject? Join the discussion and share your knowledge at Wikipedia.org.
Second derivative test
From Wikipedia, the free encyclopedia
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.
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.