# Second derivative test

From Calculus

## 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 states

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 |
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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 . |