One-sided derivative test
Statement
What the test says: one-sided sign versions
Suppose is a function and is a point in the domain of . We have the following:
| One-sided differentiability assumption | Sign assumption on one-sided derivative | Conclusion |
|---|---|---|
| is left differentiable at | The left hand derivative of at is positive | has a strict local maximum from the left at |
| is left differentiable at | The left hand derivative of at is negative | has a strict local minimum from the left at |
| is left differentiable at | The left hand derivative of at is zero | We cannot conclude anything |
| is right differentiable at | The right hand derivative of at is positive | has a strict local minimum from the right at |
| is right differentiable at | The right hand derivative of at is negative | has a strict local maximum from the right at |
| is right differentiable at | The right hand derivative of at 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 at | Sign assumption on left hand derivative of at | Sign assumption on right hand derivative of at | Conclusion for at |
|---|---|---|---|
| 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) |
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.