First derivative test and one-sided derivative test are incomparable
From Calculus
The first derivative test and one-sided derivative test are incomparable: it is possible to have situations where either test is conclusive and the other isn't.
A full table of possibilities is below.
Example function | Conclusion | Is the first derivative test conclusive? | Is direct analysis of one-sided derivatives conclusive? |
---|---|---|---|
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local minimum | Yes | No, because both one-sided derivatives are zero |
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local minimum | Yes | Yes |
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local minimum | No | Yes |
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local minimum | No | No |