One-sided version of second derivative test

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.