Second derivative test

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.