# One-sided version of second derivative test

This article describes a test that can be used to determine whether a point in the domain of a function gives a point of local, endpoint, or absolute (global) maximum or minimum of the function, and/or to narrow down the possibilities for points where such maxima or minima occur.

View a complete list of such tests

This article describes a one-sided analogue of second derivative test

## Contents

## Statement

Suppose is a function and is a point in the domain of . The **one-sided version of second derivative test** is a slight variation of the second derivative test that helps determine, using one-sided second derivatives, whether has a one-sided or two-sided local extremum at .

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

Continuity and differentiability assumption | Assumption on one-sided derivative at | Assumption on one-sided second derivative at | Conclusion about at |
---|---|---|---|

is differentiable on the immediate left of , left differentiable at and the left hand derivative function is itself left differentiable at | strict local maximum from the left | ||

is differentiable on the immediate left of , left differentiable at and the left hand derivative function is itself left differentiable at | strict local minimum from the left | ||

is differentiable on the immediate left of , left differentiable at and the left hand derivative function is itself left differentiable at | inconclusive, i.e., we may have a strict local maximum, strict local minimum, or neither from the left | ||

is differentiable on the immediate right of , right differentiable at and the right hand derivative function is itself right differentiable at | strict local maximum from the right | ||

is differentiable on the immediate right of , right differentiable at and the right hand derivative function is itself right differentiable at | strict local minimum from the right | ||

is differentiable on the immediate right of , right differentiable at and the right hand derivative function is itself right differentiable at | inconclusive, i.e., we may have a strict local maximum, strict local minimum, or neither from the right |

### What the test says: combined sign version

Note that if *either* one-sided second derivative is zero, the behavior from that side, and hence the overall behavior, is inconclusive.

Continuity and differentiability assumption | Assumption on derivative | Assumption on left second derivative | Assumption on right second derivative | Conclusion about at |
---|---|---|---|---|

is differentiable at and the left hand derivative and right hand derivative are themselves respectively left and right differentiable at | negative | negative | strict two-sided local maximum | |

is differentiable at and the left hand derivative and right hand derivative are themselves respectively left and right differentiable at | positive | positive | strict two-sided local minimum | |

is differentiable at and the left hand derivative and right hand derivative are themselves respectively left and right differentiable at | negative | positive | neither, it's a point of increase for the function | |

positive | negative | neither, it's a point of decrease for the function |

## Facts used

## Proof

### One-sided version: negative second left derivative at the point

We prove just one of the four one-sided versions with nonzero one-sided second derivatives.

**Given**: Function and point . is differentiable on the immediate left of , left differentiable at and the left hand derivative function is itself left differentiable at . and .

**To prove**: has a strict local maximum from the left at .

**Proof**:

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | The left hand derivative is defined and equal to the two-sided derivative on the immediate left of . | is differentiable on the immediate left of | The definition of left hand derivative requires the function to be defined on the immediate left. | ||

2 | The left hand derivative has a strict local minimum from the left at . | Fact (1) | . | Step (1) | We apply the appropriate subcase of Fact (1) (the one-sided derivative test) to the function at from the immediate left. |

3 | The derivative is positive for on the immediate left of . | . | Steps (1), (2) | By Step (2), is a strict local minimum from the left, which means that values on the immediate left are strictly bigger than the value at . The value equal zero, so this forces values of on the immediate left to be positive. By Step (1), is the same as on the immediate left, so we get that is positive on the immediate left of . | |

4 | has a strict local maximum from the left at . | Fact (2) | is left differentiable, hence left continuous at . | Step (3) | Follows by combining Step (3) with the appropriate one-sided case of Fact (2) (the first derivative test) and using left continuity at . |

### Combined sign version: piecing together the one-sided versions

The combined sign versions follow directly from the one-sided versions.