Second derivative test
What this test is for
What the test says
Suppose is a function and is a point in the interior of the domain of , i.e., is defined on some open interval containing . Suppose, further, that , i.e., the second derivative of , exists at . Suppose also that , so is a critical point for . Then:
|attains a local maximum value at (the value is )|
|attains a local minimum value at (the value is )|
|The test is inconclusive. may attain a local maximum value, a local minimum value, have a point of inflection, or have some different behavior at the point .|
Relation with critical points
The second derivative test is specifically used only to determine whether a critical point where the derivative is zero is a point of local maximum or local minimum. Note in particular that:
- For the other type of critical point, namely that where is undefined, the second derivative test cannot be used.
- Since point of local extremum implies critical point, we don't have to worry about points that are not critical points -- none of them will give local extrema.
Strength of the test
Second derivative test requires twice differentiability at but not around the point
The second derivative test can be applied at a critical point for a function only if is twice differentiable at . This in particular forces to be once differentiable around .
However, the test does not require to be defined around or to be continuous at .