One-sided derivative test

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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

Statement

What the test says: one-sided sign versions

Suppose f is a function and c is a point in the domain of f. We have the following:

One-sided differentiability assumption Sign assumption on one-sided derivative Conclusion
f is left differentiable at c The left hand derivative of f at c is positive f has a strict local maximum from the left at c
f is left differentiable at c The left hand derivative of f at c is negative f has a strict local minimum from the left at c
f is left differentiable at c The left hand derivative of f at c is zero We cannot conclude anything
f is right differentiable at c The right hand derivative of f at c is positive f has a strict local minimum from the right at c
f is right differentiable at c The right hand derivative of f at c is negative f has a strict local maximum from the right at c
f is right differentiable at c The right hand derivative of f at c 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 f at c Sign assumption on left hand derivative of f at c Sign assumption on right hand derivative of f at c Conclusion for f at c
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:

  1. Left hand derivative is positive implies strict local maximum from the left
  2. Left hand derivative is negative implies strict local minimum from the left
  3. Right hand derivative is positive implies strict local minimum from the right
  4. 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.