# One-sided 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

## Statement

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

Suppose $f$ is a function and $c$ is a point in the domain of $f$. We have the following:

One-sided differentiability assumption Sign assumption on one-sided derivative Conclusion
$f$ is left differentiable at $c$ The left hand derivative of $f$ at $c$ is positive $f$ has a strict local maximum from the left at $c$
$f$ is left differentiable at $c$ The left hand derivative of $f$ at $c$ is negative $f$ has a strict local minimum from the left at $c$
$f$ is left differentiable at $c$ The left hand derivative of $f$ at $c$ is zero We cannot conclude anything
$f$ is right differentiable at $c$ The right hand derivative of $f$ at $c$ is positive $f$ has a strict local minimum from the right at $c$
$f$ is right differentiable at $c$ The right hand derivative of $f$ at $c$ is negative $f$ has a strict local maximum from the right at $c$
$f$ is right differentiable at $c$ The right hand derivative of $f$ at $c$ is zero We cannot conclude anything

### What the test says: combined sign versions

Note that in the table below, we really do need strict positivity and negativity. Note that if either one-sided derivative is zero, the test is inconclusive.

One-sided differentiability assumption for $f$ at $c$ Sign assumption on left hand derivative of $f$ at $c$ Sign assumption on right hand derivative of $f$ at $c$ Conclusion for $f$ at $c$
both left and right differentiable (but not necessarily two-sided differentiable) positive negative strict local maximum
both left and right differentiable (but not necessarily two-sided differentiable) negative positive strict local minimum
both left and right differentiable (but not necessarily two-sided differentiable) positive positive neither (it's a point of increase, though the function need not increase around the point)
both left and right differentiable (but not necessarily two-sided differentiable) negative negative neither (it's a point of decrease, though the function need not decrease around the point)

## Facts used

Below are the facts for the one-sided versions:

## Proof

### One-sided sign versions

See Facts (1)-(4).

### Combined sign versions

Follows directly by suitable combinations. In particular:

• The condition for strict local maximum follows by combining (1) and (4).
• The condition for strict local minimum follows by combining (2) and (3).
• The condition for point of increase follows by combining (1) and (3).
• The condition for point of decrease follows by combining (2) and (4).

## Strength of the test

### Relation with first derivative test

For further information, refer: first derivative test and one-sided derivative test are incomparable

The one-sided derivative test is neither strictly stronger nor strictly weaker than the first derivative test. There are situations where one test works and the other doesn't, and vice versa.