One-sided version 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 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 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 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 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 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 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