The second derivative test is a method for potentially finding local extrema of a function at a critical point. The second derivative test can sometimes be inconclusive in which case the first derivative test must be used.

The second derivative test is performed by checking the second derivative at a critical point.

The test can result in 3 different cases

• If the second derivative is negative there is a local maximum
• If the second derivative is positive there is a local minimum
• If the second derivative is the test is inconclusive and the first derivative test must be used.

Example

Given the function find the local minimum(s) and local maximum(s) over the interval

is shown in blue, with critical points shown as red dots.

There are 2 local extrema, 1 local minimum at and 1 local maximum at .

Because neither of these second derivatives were 0 we know that the second derivative test was conclusive so we don't need to use the first derivative test.