# Difference between revisions of "Second derivative test"

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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. | 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 | + | ===What the test says=== |

Suppose <math>f</math> is a [[function]] and <math>c</math> is a point in the interior of the domain of <math>f</math>, i.e., <math>f</math> is defined on some [[open interval]] containing <math>c</math>. Suppose, further, that <math>f''</math>, i.e., the [[second derivative]] of <math>f</math>, exists at <math>c</math>. Suppose also that <math>f'(c)=0</math>, so <math>c</math> is a [[critical point]] for <math>f</math>. Then: | Suppose <math>f</math> is a [[function]] and <math>c</math> is a point in the interior of the domain of <math>f</math>, i.e., <math>f</math> is defined on some [[open interval]] containing <math>c</math>. Suppose, further, that <math>f''</math>, i.e., the [[second derivative]] of <math>f</math>, exists at <math>c</math>. Suppose also that <math>f'(c)=0</math>, so <math>c</math> is a [[critical point]] for <math>f</math>. Then: | ||

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| <math>f''(c) = 0</math> || The test is inconclusive. <math>f</math> may attain a local maximum value, a local minimum value, have a [[point of inflection]], or have some different behavior at the point <math>c</math>. | | <math>f''(c) = 0</math> || The test is inconclusive. <math>f</math> may attain a local maximum value, a local minimum value, have a [[point of inflection]], or have some different behavior at the point <math>c</math>. | ||

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+ | ===Relation with critical points=== | ||

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+ | 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: | ||

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+ | * For the other type of critical point, namely that where <math>f'</math> 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== | ==Related tests== | ||

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* [[First derivative test]] | * [[First derivative test]] | ||

* [[Higher derivative test]]s | * [[Higher derivative test]]s | ||

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+ | ==Strength of the test== | ||

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+ | ===Second derivative test requires twice differentiability at but not around the point=== | ||

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+ | The second derivative test can be applied at a critical point <math>c</math> for a function <math>f</math> only if <math>f</math> is ''twice'' differentiable at <math>c</math>. This in particular forces <math>f</math> to be once differentiable ''around'' <math>c</math>. | ||

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+ | However, the test does ''not'' require <math>f''</math> to be defined around <math>c</math> or to be continuous at <math>c</math>. |

## Revision as of 15:26, 2 May 2012

## 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 .