# Second derivative test

## Contents

## Statement

### What this test is for

This test is a partial test (i.e., it may be inconclusive) for determining whether a given critical point for a function is a point of local minimum, point of local maximum, or neither.

### 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:

Hypothesis | Conclusion |
---|---|

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.

## Related tests

## 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 .