One-sided derivative test
From Calculus
This article describes a test that can be used to determine whether a point in the domain of a function gives a point of local, endpoint, or absolute (global) maximum or minimum of the function, and/or to narrow down the possibilities for points where such maxima or minima occur.
View a complete list of such tests
Contents
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) |
Facts used
Below are the facts for the one-sided versions:
- Left hand derivative is positive implies strict local maximum from the left
- Left hand derivative is negative implies strict local minimum from the left
- Right hand derivative is positive implies strict local minimum from the right
- Right hand derivative is negative implies strict local maximum from the right
Proof
One-sided sign versions
See Facts (1)-(4).
Combined sign versions
Follows directly by suitable combinations. In particular:
- The condition for strict local maximum follows by combining (1) and (4).
- The condition for strict local minimum follows by combining (2) and (3).
- The condition for point of increase follows by combining (1) and (3).
- The condition for point of decrease follows by combining (2) and (4).
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.