# Difference between revisions of "First derivative test"

(→What the test says: one-sided sign versions) |
(→What the test says: one-sided sign versions) |
||

Line 7: | Line 7: | ||

===What the test says: one-sided sign versions=== | ===What the test says: one-sided sign versions=== | ||

− | Suppose <math>f</math> is a [[function]] defined | + | Suppose <math>f</math> is a [[function]] defined at a point <math>c</math>. |

Then, we have the following: | Then, we have the following: | ||

{| class="sortable" border="1" | {| class="sortable" border="1" | ||

− | ! Hypothesis !! | + | ! Hypothesis on sign of derivative!! Continuity and differentiability assumption !! Conclusion |

|- | |- | ||

− | | <math>\! f'(x)</math> is positive (respectively, nonnegative) for <math>x</math> to the immediate left of <math>c</math> (i.e., for <math>x \in (c - \delta, c)</math> for sufficiently small <math>\delta > 0</math>)|| We need <math>f</math> to be left continuous at <math>c</math> || <math>f</math> has a strict local maximum from the left at <math>c</math>, i.e., <math>f(c) > f(x)</math> (respectively, <math>f</math> has a local maximum from the left at <math>c</math>, i.e., <math>f(c) \ge f(x)</math>) for <math>x</math> to the immediate left of <math>c</math>. | + | | <math>\! f'(x)</math> is positive (respectively, nonnegative) for <math>x</math> to the immediate left of <math>c</math> (i.e., for <math>x \in (c - \delta, c)</math> for sufficiently small <math>\delta > 0</math>)|| We need <math>f</math> to be left continuous at <math>c</math> and differentiable on the immediate left of <math>c</math>. || <math>f</math> has a strict local maximum from the left at <math>c</math>, i.e., <math>f(c) > f(x)</math> (respectively, <math>f</math> has a local maximum from the left at <math>c</math>, i.e., <math>f(c) \ge f(x)</math>) for <math>x</math> to the immediate left of <math>c</math>. |

|- | |- | ||

− | | <math>\! f'(x)</math> is negative (respectively, nonpositive) for <math>x</math> to the immediate left of <math>c</math> (i.e., for <math>x \in (c - \delta, c)</math> for sufficiently small <math>\delta > 0</math>)|| We need <math>f</math> to be left continuous at <math>c</math> || <math>f</math> has a strict local minimum from the left at <math>c</math>, i.e., <math>f(c) < f(x)</math> (respectively, <math>f</math> has a local minimum from the left at <math>c</math>, i.e., <math>f(c) \le f(x)</math>) for <math>x</math> to the immediate left of <math>c</math>. | + | | <math>\! f'(x)</math> is negative (respectively, nonpositive) for <math>x</math> to the immediate left of <math>c</math> (i.e., for <math>x \in (c - \delta, c)</math> for sufficiently small <math>\delta > 0</math>)|| We need <math>f</math> to be left continuous at <math>c</math> and differentiable on the immediate left of <math>c</math>. || <math>f</math> has a strict local minimum from the left at <math>c</math>, i.e., <math>f(c) < f(x)</math> (respectively, <math>f</math> has a local minimum from the left at <math>c</math>, i.e., <math>f(c) \le f(x)</math>) for <math>x</math> to the immediate left of <math>c</math>. |

|- | |- | ||

− | | <math>\! f'(x)</math> is positive (respectively, nonnegative) for <math>x</math> to the immediate right of <math>c</math> (i.e., for <math>x \in (c,c + \delta)</math> for sufficiently small <math>\delta > 0</math>)|| We need <math>f</math> to be right continuous at <math>c</math> || <math>f</math> has a strict local minimum from the right at <math>c</math>, i.e., <math>f(c) < f(x)</math> (respectively, <math>f</math> has a local minimum from the right at <math>c</math>, i.e., <math>f(c) \le f(x)</math>) for <math>x</math> to the immediate right of <math>c</math>. | + | | <math>\! f'(x)</math> is positive (respectively, nonnegative) for <math>x</math> to the immediate right of <math>c</math> (i.e., for <math>x \in (c,c + \delta)</math> for sufficiently small <math>\delta > 0</math>)|| We need <math>f</math> to be right continuous at <math>c</math> and differentiable on the immediate right of <math>c</math> || <math>f</math> has a strict local minimum from the right at <math>c</math>, i.e., <math>f(c) < f(x)</math> (respectively, <math>f</math> has a local minimum from the right at <math>c</math>, i.e., <math>f(c) \le f(x)</math>) for <math>x</math> to the immediate right of <math>c</math>. |

|- | |- | ||

− | | <math>\! f'(x)</math> is negative (respectively, nonpositive) for <math>x</math> to the immediate right of <math>c</math> (i.e., for <math>x \in (c,c + \delta)</math> for sufficiently small <math>\delta > 0</math>)|| We need <math>f</math> to be right continuous at <math>c</math> || <math>f</math> has a strict local maximum from the right at <math>c</math>, i.e., <math>f(c) > f(x)</math> (respectively, <math>f</math> has a local maximum from the right at <math>c</math>, i.e., <math>f(c) \ge f(x)</math>) for <math>x</math> to the immediate right of <math>c</math>. | + | | <math>\! f'(x)</math> is negative (respectively, nonpositive) for <math>x</math> to the immediate right of <math>c</math> (i.e., for <math>x \in (c,c + \delta)</math> for sufficiently small <math>\delta > 0</math>)|| We need <math>f</math> to be right continuous at <math>c</math> and differentiable on the immediate right of <math>c</math> || <math>f</math> has a strict local maximum from the right at <math>c</math>, i.e., <math>f(c) > f(x)</math> (respectively, <math>f</math> has a local maximum from the right at <math>c</math>, i.e., <math>f(c) \ge f(x)</math>) for <math>x</math> to the immediate right of <math>c</math>. |

|} | |} | ||

## Revision as of 16:06, 24 April 2012

## Contents

## Statement

### What the test is for

The **first derivative test** is a partial (i.e., not always conclusive) test used to determine whether a particular critical point in the domain of a function is a point where the function attains a local maximum value, local minimum value, or neither. There are cases where the test is *inconclusive*, which means that we cannot draw any conclusion.

### What the test says: one-sided sign versions

Suppose is a function defined at a point .

Then, we have the following:

Hypothesis on sign of derivative | Continuity and differentiability assumption | Conclusion |
---|---|---|

is positive (respectively, nonnegative) for to the immediate left of (i.e., for for sufficiently small ) | We need to be left continuous at and differentiable on the immediate left of . | has a strict local maximum from the left at , i.e., (respectively, has a local maximum from the left at , i.e., ) for to the immediate left of . |

is negative (respectively, nonpositive) for to the immediate left of (i.e., for for sufficiently small ) | We need to be left continuous at and differentiable on the immediate left of . | has a strict local minimum from the left at , i.e., (respectively, has a local minimum from the left at , i.e., ) for to the immediate left of . |

is positive (respectively, nonnegative) for to the immediate right of (i.e., for for sufficiently small ) | We need to be right continuous at and differentiable on the immediate right of | has a strict local minimum from the right at , i.e., (respectively, has a local minimum from the right at , i.e., ) for to the immediate right of . |

is negative (respectively, nonpositive) for to the immediate right of (i.e., for for sufficiently small ) | We need to be right continuous at and differentiable on the immediate right of | has a strict local maximum from the right at , i.e., (respectively, has a local maximum from the right at , i.e., ) for to the immediate right of . |

### What the test says: combined sign versions

Suppose is a function defined around a point (i.e., is defined in an open interval containing ) and is continuous at (What this means is that we do not care whether is differentiable at ; however, it must be continuous at and differentiable at points to the immediate left and immediate right of ).

Then, we have the following (we list only the *strict* cases in the table below):

Sign of on immediate left of | Sign of on immediate right of | Conclusion about local minimum, local maximum, or neither |
---|---|---|

positive | negative | strict local maximum |

positive | positive | neither local maximum nor local minimum |

negative | negative | neither local maximum nor local minimum |

negative | positive | strict local minimum |

If we replace *positive* by *nonnegative* and *negative* by *nonpositive* in the rows corresponding to strict local maximum and strict local minimum, we could potentially lose the *strictness*.

### Succinct version

Here is a shorter version: at a critical point, if the derivative changes sign from negative to positive (as we go from left to right) then that is a point of local minimum. If the derivative changes sign from positive to negative (as we go from left to right) then it is a point of local maximum.

## Related tests

## Notes

### Points of inflection

### Examples illustrating why the test is not always conclusive

The following problems could occur when applying this test:

- The function is not continuous, or not differentiable, at points to the immediate left or immediate right of the critical point.
- The function is differentiable on the immediate left and immediate right of the critical point. However, the derivative does not have a uniform sign on the immediate left or the immediate right, i.e., it is oscillatory in sign at points arbitrarily close to the critical point.

Here is a picture of a function illustrating (2):