First derivative test is inconclusive for function whose derivative has ambiguous sign around the point
This article describes a situation, or broad range of situations, where a particular test or criterion is inconclusive, i.e., it does not work as intended to help us determine what we would like to determine. This is either because one of the hypotheses for the test fails or we land up in an inconclusive branch of the test.
The test is first derivative test. See more inconclusive cases for first derivative test | conclusive cases for first derivative test
Contents
Statement
Goal of statement
The goal of this statement is to identify a type of situation where the first derivative test is inconclusive.
One-sided version
Suppose is a real-valued function of one variable and
is a point in the domain of
.
Continuity and differentiability assumption | Hypothesis on sign of derivative ![]() |
Conclusion |
---|---|---|
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Two-sided version
Suppose is a real-valued function of one variable and
is a point in the domain of
.
Continuity and differentiability assumption | Sign of ![]() |
Sign of ![]() |
Conclusion |
---|---|---|---|
![]() ![]() ![]() |
oscillatory | oscillatory | ![]() |
![]() ![]() ![]() |
oscillatory | positive | ![]() ![]() |
![]() ![]() ![]() |
oscillatory | negative | ![]() ![]() |
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positive | oscillatory | ![]() ![]() |
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negative | oscillatory | ![]() ![]() |
Relation with critical points
In all the two-sided cases, the point under consideration must be a critical point for the function (i.e., either
or
does not exist). Thus, we could add, to each case, the additional condition that
be a critical point for the function, and this will not affect the strength of the statements.
Example
Example of two-sided local minimum despite oscillatory sign of derivative around the point
The function illustrated in the picture is:
We note the following:
Assertion | Explanation |
---|---|
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For ![]() ![]() ![]() ![]() |
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We can see this using the pinching theorem (by pinching the function between ![]() ![]() ![]() ![]() ![]() |
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For ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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For ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Example of oscillatory sign of derivative where the function does not have a local extremum from either side
An example is:
Assertion | Construction |
---|---|
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This is because ![]() ![]() ![]() ![]() |
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We can see this using the pinching theorem or more directly by noting that as ![]() ![]() ![]() |
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The derivative is ![]() ![]() ![]() ![]() ![]() |
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The derivative is ![]() ![]() ![]() ![]() ![]() |
Variations that cover all cases
The above two examples can be modified to produce examples for all the cases mentioned in the statement. To convert local minimum to local maximum, multiply the whole function by . To obtain one-sided behavior, restrict the analysis to one side. Also, use piecewise combinations with functions that are nice on the other side to obtain other examples for two-sided behavior.
In all cases, we take the point and the value
for convenience in all examples. This allows us to easily make two-sided combinations.
One-sided requirement | Example function (definition only on appropriate side) |
---|---|
Oscillatory sign of derivative on left, but function has strict local maximum from left | ![]() |
Oscillatory sign of derivative on left, but function has strict local minimum from left | ![]() |
Oscillatory sign of derivative on left, and function has neither local maximum nor local minimum from left | ![]() |
Oscillatory sign of derivative on right, but function has strict local maximum from right | ![]() |
Oscillatory sign of derivative on right, but function has strict local minimum from right | ![]() |
Oscillatory sign of derivative on right, and function has neither local maximum nor local minimum from right | ![]() |