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 $c$ .	The first $k-1$ left hand derivatives at $c$ 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 $c$ .	The first $k-1$ left hand derivatives at $c$ 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 $c$ .	The first $k-1$ left hand derivatives at $c$ 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 $c$ .	The first $k-1$ left hand derivatives at $c$ 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 $c$ .	The first $k-1$ right hand derivatives at $c$ 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 $c$ .	The first $k-1$ right hand derivatives at $c$ 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 on left and right

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.	$f(x):=x\|x\|,c=0$
$f$ 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	$f(x):=x^{2}\|x\|,c=0$
$f$ 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.	$f(x):=-x\|x\|,c=0$
$f$ 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	$f(x):=-x^{2}\|x\|,c=0$
$f$ 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.	$f(x):=x^{3},c=0$

Facts used

Proof

Proof idea

We prove by induction on $k$ , so we assume that the result holds for $k-1$ and then prove it for $k$ .

The base case of induction is Fact (1), the one-sided version of second derivative test.

The way we use the inductive hypothesis is that we apply it, not to the original function $f$ , but to the derivative $f'_{\pm }$ . We combine the inductive hypothesis with the fact that the derivative value is zero to conclude something about its sign on the immediate left or right, then we use Fact (2), the first derivative test.

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