One-sided version of higher 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 higher derivative test

Statement

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

We denote by $f_{-}^{(r)}(c)$ the function obtained by iterating the left hand derivative operation on $f$ $r$ times at the point $c$. Similarly, we denote by $f_{+}^{(r)}(c)$ the function obtained by iterating the right hand derivative operation on $f$ $r$ times at the point $c$.

What the test says: one-sided sign version

In the table below, $k$ is a positive integer greater than 1.

Note that for the left side approach, the parity of $k$ (even versus odd) matters for the conclusion, whereas for the right side approach, the parity of $k$ does not matter.

Continuity and differentiability assumption	Assumption on one-sided derivative values $f_{\pm }^{(r)}(c),r<k$	Assumption on $f_{\pm }^{(k)}(c)$	Assumption on parity of $k$	Conclusion about one-sided local extremum of $f$ at $c$	Prototypical example
$f$ is left continuous and (at least) $k$ times left differentiable at $c$. Further, $f$ is $(k-1)$ times (two-sided) differentiable on the immediate left of $c$.	All of them are equal to zero	$f_{-}^{(k)}(c)$ is negative	even	$f$ has strict local maximum from the left at $c$.	$f(x):=-x^2,c=0$. Here, $k=2,f^{(k)}(c)=-2$.
$f$ is left continuous and (at least) $k$ times left differentiable at $c$. Further, $f$ is $(k-1)$ times (two-sided) differentiable on the immediate left of $c$.	All of them are equal to zero	$f_{-}^{(k)}(c)$ is negative	odd	$f$ has strict local minimum from the left at $c$.	$f(x):=-x^3,c=0$. Here, $k=3,f^{(k)}(c)=-6$.
$f$ is left continuous and (at least) $k$ times left differentiable at $c$. Further, $f$ is $(k-1)$ times (two-sided) differentiable on the immediate left of $c$.	All of them are equal to zero	$f_{-}^{(k)}(c)$ is positive	even	$f$ has strict local minimum from the left at $c$.	$f(x):=x^2,c=0$
$f$ is left continuous and (at least) $k$ times left differentiable at $c$. Further, $f$ is $(k-1)$ times (two-sided) differentiable on the immediate left of $c$.	All of them are equal to zero	$f_{-}^{(k)}(c)$ is positive	odd	$f$ has strict local maximum from the left at $c$.	$f(x):=x^3,c=0$
$f$ is right continuous and (at least) $k$ times right differentiable at $c$. Further, $f$ is $(k-1)$ times (two-sided) differentiable on the immediate right of $c$.	All of them are equal to zero	$f_{+}^{(k)}(c)$ is negative	doesn't matter	$f$ has strict local maximum from the right at $c$.	$f(x):=-x^2,c=0$ or $f(x):=-x^3,c=0$
$f$ is right continuous and (at least) $k$ times right differentiable at $c$. Further, $f$ is $(k-1)$ times (two-sided) differentiable on the immediate right of $c$.	All of them are equal to zero	$f_{+}^{(k)}(c)$ is positive	doesn't matter	$f$ has strict local minimum from the right at $c$.	$f(x):=x^2,c=0$ or $f(x):=x^3,c=0$

What the test says: combined sign version with same position of first nonzero derivative

In the table below, $k$ is a positive integer greater than 1. Note that when we say $f$ is differentiable a certain number of times, we mean it is differentiable at least that many times.

Continuity and differentiability assumption	Assumption on derivative values $f^{(r)}(c),r<k$	Assumption on $f_{-}^{(k)}(c)$	Assumption on $f_{+}^{(k)}(c)$	Assumption on parity of $k$	Conclusion about two-sided local extremum of $f$ at $c$	Prototypical example
$f$ is $(k-1)$ times differentiable at and around $c$, and $k$ times left and right differentiable at $c$.	All of them are equal to zero	negative	negative	even	strict local maximum	$f(x):=-x^2,c=0$
$f$ is $(k-1)$ times differentiable at and around $c$, and $k$ times left and right differentiable at $c$.	All of them are equal to zero	negative	negative	odd	neither local maximum nor local minimum. The function decreases through the point.	$f(x):=-x^3,c=0$
$f$ is $(k-1)$ times differentiable at and around $c$, and $k$ times left and right differentiable at $c$.	All of them are equal to zero	negative	positive	even	neither local maximum nor local minimum. The function increases through the point.	$f(x):=x\left|x\right|,c=0$
$f$ is $(k-1)$ times differentiable at and around $c$, and $k$ times left and right differentiable at $c$.	All of them are equal to zero	negative	positive	odd	strict local minimum	$f(x):=x^2\left|x\right|,c=0$
$f$ is $(k-1)$ times differentiable at and around $c$, and $k$ times left and right differentiable at $c$.	All of them are equal to zero	positive	negative	even	neither local maximum nor local minimum. The function decreases through the point.	$f(x):=-x\left|x\right|,c=0$
$f$ is $(k-1)$ times differentiable at and around $c$, and $k$ times left and right differentiable at $c$.	All of them are equal to zero	positive	negative	odd	strict local maximum	$f(x):=-x^2\left|x\right|,c=0$
$f$ is $(k-1)$ times differentiable at and around $c$, and $k$ times left and right differentiable at $c$.	All of them are equal to zero	positive	positive	even	strict local minimum	$f(x):=x^2,c=0$
$f$ is $(k-1)$ times differentiable at and around $c$, and $k$ times left and right differentiable at $c$.	All of them are equal to zero	positive	positive	odd	neither local maximum nor local minimum. The function increases through the point.	$f(x):=x^3,c=0$