SECOND DERIVATIVE TEST

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 neighborhood of a stationary point $x$, then:

★ If then $f$ has a maximum at $x$.

★ If $f^\{primeprime\}(x)\; >\; 0$ then $f$ has a minimum at $x$.

★ If $f^\{primeprime\}(x)\; =\; 0$, the second derivative test says nothing about the point $x$.

Contents

Multivariable case

Concavity test

See also

References

Multivariable case

Main articles: 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 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.

Concavity test

The second derivative test may also be used to determine the concavity of a function as well as a function's points of inflection.
Firstly, all points at which $f\text{'}(x)\; =\; 0$ are found. In each of the intervals created, $f\text{'}\text{'}(x)$ is then evaluated at a single point. For the intervals where the evaluated value of the function $f(x)$ is concave down, and for all intervals between critical points where the evaluated value of $f\text{'}\text{'}(x)\; >\; 0$ the function $f(x)$ is concave up. The points that separate intervals of opposing concavity are points of inflection.

See also

★ Fermat's theorem

★ First derivative test

★ Higher order derivative test

★ Differentiability

★ Extreme value

★ Inflection point

★ Convex function

★ Concave function

References

★ Second Derivative Test from 'Mathworld'

★ Concavity and the Second Derivative Test

★ Thomas Simpson's use of Second Derivative Test to Find Maxima and Minima at Convergence