# First derivative test

## 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:

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

is left continuous at and differentiable on the immediate left of |
is positive (respectively, nonnegative) for to the immediate left of (i.e., for for sufficiently small ) | 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 left continuous at and differentiable on the immediate left of |
is negative (respectively, nonpositive) for to the immediate left of (i.e., for for sufficiently small ) | 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 right continuous at and differentiable on the immediate right of |
is positive (respectively, nonnegative) for to the immediate right of (i.e., for for sufficiently small ) | 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 right continuous at and differentiable on the immediate right of |
is negative (respectively, nonpositive) for to the immediate right of (i.e., for for sufficiently small ) | 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 . We do not care whether is differentiable at ; however, the test makes sense only if is differentiable on the immediate left and immediate right of .

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

Continuity and differentiability assumption | Sign of the derivative on immediate left of | Sign of on immediate right of | Conclusion about local minimum, local maximum, or neither |
---|---|---|---|

is continuous at and differentiable on the immediate left and immediate right of | positive | negative | strict local maximum |

is continuous at and differentiable on the immediate left and immediate right of | negative | positive | strict local minimum |

is continuous at and differentiable on the immediate left and immediate right of | positive | positive | neither local maximum nor local minimum, because is increasing through the point |

is continuous at and differentiable on the immediate left and immediate right of | negative | negative | neither local maximum nor local minimum, because is decreasing through the point |

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*.

Note that if has ambiguous sign on the immediate left or on the immediate right of , the first derivative test is inconclusive.

### Relation with critical points

The typical goal of the first derivative test is to determine whether a critical point is a point of local maximum or minimum. Hence, the test is typically applied to critical points. *However, when applying the first derivative test, we do not need to check whether the point in question is a critical point. In other words, if the condition for being a point of local maximum or minimum is satisfied, then the point in question is automatically a critical point and this condition need not be checked separately.*

### 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

### First derivative test does not require differentiability at the point

To apply the two-sided combined sign version of the first derivative test, we need *continuity* at the point and differentiability on the immediate left and immediate right of the point. However, we do not require differentiability *at* the point.

Thus, for instance, the first derivative test can be used to study the behavior of a function with a piecewise definition by interval, such that the function is changing definition at the point. Explicitly, it can be used to study functions of the form:

Assume that is continuous at , i.e., . In that case, we can try to determine whether is a point of local maximum, minimum, or neither by studying the sign of to the immediate left of and the sign of to the immediate right of . It is not necessary that be differentiable at (for more on how to differentiate piecewise functions, see differentiation rule for piecewise definition by interval).

### Situations where the test is not conclusive

The following problems could occur when applying this test:

- First derivative test fails for function that is discontinuous at the critical point: If the function is not continuous at the critical point, then the first derivative test may yield incorrect conclusions.
- The first derivative test fails (or rather, cannot be applied) if the function is not differentiable on the immediate left or immediate right of the point.
- First derivative test is inconclusive for function whose derivative has ambiguous sign around the point: A pictorial illustration is below:

### Condition for the test to be conclusive

- First derivative test is conclusive for differentiable function at isolated critical point: If is differentiable on the immediate left and immediate right of a critical point ,
*and*is an isolated critical point (i.e., there is an open interval containing it that contains no other critical points), then the first derivative test must be conclusive at . In other words, we can use the first derivative test to definitively determine whether is a point of local maximum, local minimum, or neither, for . - In particular, the first derivative test is always conclusive for polynomials and rational functions. It is also conclusive for functions with piecewise definition by interval where each of the piece is a polynomial or rational function.