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 |
---|---|---|
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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 ![]() ![]() |
Sign of ![]() ![]() |
Conclusion about local minimum, local maximum, or neither |
---|---|---|---|
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positive | negative | strict local maximum |
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negative | positive | strict local minimum |
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positive | positive | neither local maximum nor local minimum, because ![]() |
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negative | negative | neither local maximum nor local minimum, because ![]() |
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).
Examples illustrating why the test is not always 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: