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In calculus, a branch of mathematics, the second derivative test is a criterion often useful for determining whether a given stationary point of a function is a local maximum or a local minimum.## 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.
## Proof of Second Derivative Test

## Concavity test

## See also

## References

The test states: If the function $f$ is twice differentiable in a neighborhood of a stationary point $x$, meaning that $f^\{prime\}(x)\; =\; 0$, then:

- If $f^\{primeprime\}(x)\; <\; 0$ then $f$ has a local maximum at $x$.
- If $f^\{primeprime\}(x)\; >\; 0$ then $f$ has a local minimum at $x$.
- If $f^\{primeprime\}(x)\; =\; 0$, the second derivative test says nothing about the point $x$.

In the last case, the function may have a local maximum or minimum there, but the function is sufficiently "flat" that this is undetected by the second derivative. Such an example is $f(x)\; =\; x\; ^4$.

Suppose we have $f$(x) > 0 (the proof for $f$(x) < 0 is analogous). Then

- $0\; <\; f(x)\; =\; lim\_\{h\; to\; 0\}\; frac\{f\text{'}(x\; +\; h)\; -\; f\text{'}(x)\}\{h\}\; =\; lim\_\{h\; to\; 0\}\; frac\{f\text{'}(x\; +\; h)\; -\; 0\}\{h\}\; =\; lim\_\{h\; to\; 0\}\; frac\{f\text{'}(x+h)\}\{h\}$

Thus, for h sufficiently small we get

- $frac\{f\text{'}(x+h)\}\{h\}\; >\; 0$

- $f\text{'}(x+h)\; <\; 0$ if h < 0, and

- $f\text{'}(x+h)\; >\; 0$ if h > 0.

Now, by the first derivative test we know that $f$ has a local minimum at $x$.

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

- Fermat's theorem
- First derivative test
- Higher order derivative test
- Differentiability
- Extreme value
- Inflection point
- Convex function
- Concave function

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Last updated on Saturday June 21, 2008 at 19:57:05 PDT (GMT -0700)

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Last updated on Saturday June 21, 2008 at 19:57:05 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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