Definitions

# Second derivative test

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.

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

• If $f^\left\{primeprime\right\}\left(x\right) < 0$ then $f$ has a local maximum at $x$.
• If $f^\left\{primeprime\right\}\left(x\right) > 0$ then $f$ has a local minimum at $x$.
• If $f^\left\{primeprime\right\}\left(x\right) = 0$, 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 $f\left(x\right) = x ^4$.

## 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

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

$0 < f\left(x\right) = lim_\left\{h to 0\right\} frac\left\{f\text{'}\left(x + h\right) - f\text{'}\left(x\right)\right\}\left\{h\right\} = lim_\left\{h to 0\right\} frac\left\{f\text{'}\left(x + h\right) - 0\right\}\left\{h\right\} = lim_\left\{h to 0\right\} frac\left\{f\text{'}\left(x+h\right)\right\}\left\{h\right\}$

Thus, for h sufficiently small we get

$frac\left\{f\text{'}\left(x+h\right)\right\}\left\{h\right\} > 0$
which means that
$f\text{'}\left(x+h\right) < 0$ if h < 0, and
$f\text{'}\left(x+h\right) > 0$ if h > 0.

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

## 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. First, all points at which $f\text{'}\left(x\right) = 0$ 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 $f\left(x\right)$ is concave down, and for all intervals between critical points where the evaluated value of $f\left(x\right) > 0$ the function $f\left(x\right)$ is concave up. The points that separate intervals of opposing concavity are points of inflection.

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