Second derivative test

Second derivative test
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "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 $\ f^{\prime}(x) = 0$ , then:

If $\ f^{\prime\prime}(x) < 0$ then $\ f$ has a local maximum at $\ x$ .
If $\ f^{\prime\prime}(x) > 0$ then $\ f$ has a local minimum at $\ x$ .
If $\ f^{\prime\prime}(x) = 0$ , 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. In this case one has to examine the third derivative. Such an example is $f (x) = x 4$ .

content

1 Multivariable case
2 Proof of Second Derivative Test
3 Concavity test
4 See also
5 References

Multivariable case

Main article: 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.

Proof of Second Derivative Test

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

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

Thus, for h sufficiently small we get

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

which means that

f'(x + h) < 0

if h < 0, and

f'(x + h) > 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 $\ f'(x) = 0$ 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.

References

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Contents

Multivariable case

Proof of Second Derivative Test

Concavity test

See also

References