, 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 is twice differentiable in a neighborhood of a stationary point , meaning that
- If then has a local maximum at .
- If then has a local minimum at .
- If , the second derivative test says nothing about the point .
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 .
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 (the proof for is analogous). Then
Thus, for h sufficiently small we get
which means that
- if h < 0, and
- if h > 0.
Now, by the first derivative test we know that has a local minimum at .
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 are found. In each of the intervals created, is then evaluated at a single point. For the intervals where the evaluated value of the function is concave down, and for all intervals between critical points where the evaluated value of the function is concave up. The points that separate intervals of opposing concavity are points of inflection.