Second derivative test

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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.

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. Such an example is $f (x) = x 4$ .

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

[edit] 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.

[edit] Proof of Second Derivative Test

0<ƒ”(x) = lim h→ 0 (ƒ’(x+h)-ƒ’(x))/h = lim h→ 0 (ƒ’(x+h)-0)/h = (ƒ’(x+h))/h

Thus, for h sufficiently small we get

(ƒ’(x+h))/h > 0

Considering h significantly small we get

If h<0, ƒ’(x+h) < 0 and if h> 0, ƒ’ (x+h) > 0.

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

Proof for 0 >ƒ”(x) follows from proof above.

[edit] 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.