One-sided version of second 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
This article describes a one-sided analogue of second derivative test

Statement

Suppose f is a function and c is a point in the domain of f. The one-sided version of second derivative test is a slight variation of the second derivative test that helps determine, using one-sided second derivatives, whether f has a one-sided or two-sided local extremum at c.

What the test says: one-sided sign version

Continuity and differentiability assumption Assumption on one-sided derivative at c Assumption on one-sided second derivative at c Conclusion about f at c
f is differentiable on the immediate left of c, left differentiable at c and the left hand derivative function is itself left differentiable at c f'_-(c) = 0 (f'_-)'_-(c) < 0 strict local maximum from the left
f is differentiable on the immediate left of c, left differentiable at c and the left hand derivative function is itself left differentiable at c f'_-(c) = 0 (f'_-)'_-(c) > 0 strict local minimum from the left
f is differentiable on the immediate left of c, left differentiable at c and the left hand derivative function is itself left differentiable at c f'_-(c) = 0 (f'_-)'_-(c) = 0 inconclusive, i.e., we may have a strict local maximum, strict local minimum, or neither from the left
f is differentiable on the immediate right of c, right differentiable at c and the right hand derivative function is itself right differentiable at c f'_+(c) = 0 (f'_+)'_+(c) < 0 strict local maximum from the right
f is differentiable on the immediate right of c, right differentiable at c and the right hand derivative function is itself right differentiable at c f'_+(c) = 0 (f'_+)'_+(c) > 0 strict local minimum from the right
f is differentiable on the immediate right of c, right differentiable at c and the right hand derivative function is itself right differentiable at c f'_+(c) = 0 (f'_+)'_+(c) = 0 inconclusive, i.e., we may have a strict local maximum, strict local minimum, or neither from the right

What the test says: combined sign version

Note that if either one-sided second derivative is zero, the behavior from that side, and hence the overall behavior, is inconclusive.

Continuity and differentiability assumption Assumption on derivative Assumption on left second derivative (f'_-)'_-(c) Assumption on right second derivative (f'_+)'_+(c) Conclusion about f at c
f is differentiable at c and the left hand derivative and right hand derivative are themselves respectively left and right differentiable at c f'(c) = 0 negative negative strict two-sided local maximum
f is differentiable at c and the left hand derivative and right hand derivative are themselves respectively left and right differentiable at c f'(c) = 0 positive positive strict two-sided local minimum
f is differentiable at c and the left hand derivative and right hand derivative are themselves respectively left and right differentiable at c f'(c) = 0 negative positive neither, it's a point of increase for the function
f is differentiable at c and the left hand derivative and right hand derivative are themselves respectively left and right differentiable at c f'(c) = 0 positive negative neither, it's a point of decrease for the function

Facts used

  1. One-sided derivative test
  2. First derivative test

Proof

One-sided version: negative second left derivative at the point

We prove just one of the four one-sided versions with nonzero one-sided second derivatives.

Given: Function f and point c. f is differentiable on the immediate left of c, left differentiable at c and the left hand derivative function is itself left differentiable at c. f'_-(c) = 0 and (f'_-)'_-(c) < 0.

To prove: f has a strict local maximum from the left at c.

Proof:

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 The left hand derivative f'_- is defined and equal to the two-sided derivative f' on the immediate left of c. f is differentiable on the immediate left of c The definition of left hand derivative requires the function to be defined on the immediate left.
2 The left hand derivative f'_- has a strict local minimum from the left at c. Fact (1) (f'_-)'_-(c) < 0. Step (1) We apply the appropriate subcase of Fact (1) (the one-sided derivative test) to the function f'_- at c from the immediate left.
3 The derivative f' is positive for x on the immediate left of c. f'_-(c) = 0. Steps (1), (2) By Step (2), f'_-(c) is a strict local minimum from the left, which means that values on the immediate left are strictly bigger than the value at c. The value f'_-(c) equal zero, so this forces values of f'_- on the immediate left to be positive. By Step (1), f'_- is the same as f' on the immediate left, so we get that f' is positive on the immediate left of c.
4 f has a strict local maximum from the left at c. Fact (2) f is left differentiable, hence left continuous at c. Step (3) Follows by combining Step (3) with the appropriate one-sided case of Fact (2) (the first derivative test) and using left continuity at c.

Combined sign version: piecing together the one-sided versions

The combined sign versions follow directly from the one-sided versions.