# First derivative test and one-sided derivative test are incomparable

From Calculus

The first derivative test and one-sided derivative test are incomparable: it is possible to have situations where either test is conclusive and the other isn't.

A full table of possibilities is below.

Example function | Conclusion | Is the first derivative test conclusive? | Is direct analysis of one-sided derivatives conclusive? |
---|---|---|---|

, | local minimum | Yes | No, because both one-sided derivatives are zero |

, | local minimum | Yes | Yes |

, | local minimum | No | Yes |

, | local minimum | No | No |