Second derivative test
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In calculus, a branch of mathematics, the second derivative test determines whether a given stationary point of a function (where its first derivative is zero) is a maximum, a minimum, or neither.
The first derivative test relates the condition of being a maximum or a minimum to a condition on the positivity or negativity of the first derivative. The second derivative test works by rephrasing the condition on the first derivative in terms of the second derivative. Suppose that f is twice differentiable in a neighbourhood of a stationary point x. The test says:
- If there exists a positive number r such that f'' is continuous between x-r and x+r, and if f''(x) is positive, then f has a minimum at x.
- If there exists a positive number r such that f'' is continuous between x-r and x+r, and if f''(x) is negative, then f has a maximum at x.
- If f'' is not continuous between x-r and x+r for any r, or if for some r, f'' is continuous between x-r and x+r but f''(x) is zero, then the test fails.
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 question. 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 in inconclusive.
See also
- First derivative test
- Higher order derivative test
- Differentiability
- Extreme value
- Inflection point
- Convex function
- Concave function