Second derivative test
In calculus, the second derivative test is a criterion for determining whether a given critical point of a real function of one variable is a local maximum or a local minimum using the value of the second derivative at the point.
The test states: if the function f is twice differentiable at a critical point x (i.e. f'(x) = 0), then:
- If then has a local maximum at .
- If then has a local minimum at .
- If , the test is inconclusive.
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 critical point. In particular, assuming that all second order partial derivatives of f are continuous on a neighbourhood of a critical 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 the second derivative test
Suppose we have (the proof for is analogous). By assumption, . Then
Thus, for h sufficiently small we get
which means that if h < 0 (intuitively, f is decreasing as it approaches x from the left), and that if h > 0 (intuitively, f is increasing as we go right from x). Now, by the first derivative test, has a local minimum at .
A related but distinct use of second derivatives is to determine whether a function is concave up or concave down at a point. It does not, however, provide information about inflection points. Specifically, a twice-differentiable function f is concave up if and concave down if . Note that if , then has zero second derivative, yet is not an inflection point, so the second derivative alone does not give enough information to determine if a given point is an inflection point.
- Bordered Hessian
- First derivative test
- Optimization (mathematics)
- Fermat's theorem
- Higher-order derivative test
- Extreme value
- Convex function