Point of local extremum implies critical point

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

Statement

Suppose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is a function of one variable and $c$ is a point in the interior of the domain of $f$ (i.e., $f$ is defined on an open interval containing $c$).

Suppose further that $c$ is a point of local extremum for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} , i.e., $f$ attains a local extreme value (either a local maximum or a local minimum) at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} .

Then, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} is a critical point for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} , i.e., either the derivative Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \! f'(c)} equals zero or the derivative Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \! f'(c)} does not exist.

Note that the $\!f'(c)$ not existing case could occur in either of these ways: one or both the one-sided derivatives of $f$ at $c$ not existing, or both one-sided derivatives of $f$ at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} existing but being unequal.

Facts used

Statement of facts

Fact no. (for reference in proof)	Statement	Assumption about one-sided local extremum	Conclusion about sign of one-sided derivative	Quick explanation in terms of the difference quotient ${\frac {f(x)-f(c)}{x-c}}$	Prototypical pictures
1	Local maximum from the left implies left hand derivative is nonnegative if it exists (has full proof + video)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} has a (possibly, but not necessarily, strict) local maximum from the left at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} .	The left hand derivative $f'_{-}(c)$, if it exists, is positive or zero	In the difference quotient, the numerator is negative or zero and the denominator is negative. The quotient is thus positive or zero. Hence, so is the limit, if it exists.
2	Local minimum from the left implies left hand derivative is nonpositive if it exists	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} has a (possibly, but not necessarily, strict) local minimum from the left at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} .	The left hand derivative Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'_-(c)} , if it exists, is negative or zero	In the difference quotient, the numerator is positive or zero and the denominator is negative. The quotient is thus negative or zero. Hence, so is the limit, if it exists.
3	Local maximum from the right implies right hand derivative is nonpositive if it exists	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} has a (possibly, but not necessarily, strict) local maximum from the right at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} .	The right hand derivative $f'_{+}(c)$, if it exists, is negative or zero	In the difference quotient, the numerator is negative or zero and the denominator is positive. The quotient is thus negative or zero. Hence, so is the limit, if it exists.
4	Local minimum from the right implies right hand derivative is nonnegative if it exists	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} has a (possibly, but not necessarily, strict) local minimum from the right at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} .	The right hand derivative $f'_{+}(c)$, if it exists, is positive or zero	In the difference quotient, the numerator is positive or zero and the denominator is positive. The quotient is thus positive or zero. Hence, so is the limit, if it exists.

The video below provides an intuitive explanation of the above facts. For a full proof, see the page on Fact (1).

{{#widget:YouTube|id=55MZGhtagFg}}

Note on strictness

All the facts as stated above are for not necessarily strict one-sided local extrema, i.e., we allow $f(x)=f(c)$ for $x$ on the immediate left or right of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} . However, even if we impose the additional condition of strictness, we still cannot eliminate the possibility of the one-sided derivative being zero. The reason is that even though the difference quotient must now be strictly positive or strictly negative in the various cases, the one-sided derivative, which is defined as a one-sided limit of the difference quotient, may still be equal to zero. All the prototypical pictures in the previous section are pictures of strict local extrema, and some of them show a one-sided derivative of zero.

For more, see the note on strictness in the proof for Fact (1).

This observation will be crucial when we piece together the two-sided information.

Note on sign sensitivity

In the facts used above, we see that, when ascertaining the sign of the difference quotient ${\frac {f(x)-f(c)}{x-c}}$, the following are true:

• The sign of the numerator is governed by whether the point is a point of local maximum or minimum. For a point of local maximum, the numerator is negative or zero, and for a point of local minimum, the numerator is positive or zero.
• The sign of the denominator is governed by whether we are approaching from the left or the right. For a left-sided approach, the denominator is negative and for a right-sided approach, the denominator is positive.

The upshot of this is that if we change the direction of approach while preserving the nature of the local extreme value, the sign of the one-sided derivative flips. This is crucial to the proof.

Proof

Proof idea

The idea is to convert two-sided local extremum information into both its one-sided pieces, then determine the signs of the one-sided derivatives. As noted in #Note on sign sensitivity, the sign conclusions for the two one-sided derivatives are opposite. We then pit these two pieces of information against each other to force the two-sided derivative, if it exists, to equal zero.

Local maximum case

Given: A function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} , a point $c$ in the interior of the domain of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} such that $f$ attains a local maximum at $c$, i.e., $f(x)\leq f(c)$ for all $x\in (c-\delta ,c+\delta )$ for some choice of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta > 0} .

To prove: If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \! f'(c)} exists, then $\!f'(c)=0$

Proof

Step no.	Assertion/construction	Facts used	Given data used	Previous steps used	Explanation
1	If the left hand derivative of $f$ at $c$ exists, then it is nonnegative, i.e., it is positive or zero.	Fact (1)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} attains a (two-sided) local maximum at $c$		Since $f$ attains a two-sided local maximum at $c$, it in particular attains a local maximum from the left. Thus, Fact (1) applies.
2	If the right hand derivative of $f$ at $c$ exists, then it is nonpositive, i.e., it is negative or zero.	Fact (3)	$f$ attains a (two-sided) local maximum at $c$		Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} attains a two-sided local maximum at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} , it in particular attains a local maximum from the right. Thus, Fact (3) applies.
3	If the (two-sided) derivative of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} exists at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} , it must be zero.			Steps (1), (2)	For the (two-sided) derivative to exist, both the left hand derivative and the right hand derivative must exist and they must be equal to each other and to the derivative. By Step (1), the derivative must be positive or zero. By Step (2), the derivative must be negative or zero. The only way both of these can be simultaneously true is if the derivative equals zero.

{{#widget:YouTube|id=UWo7nG9ufGM}}

Local minimum case

Given: A function $f$, a point $c$ in the interior of the domain of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} attains a local minimum at $c$, i.e., $f(x)\geq f(c)$ for all $x\in (c-\delta ,c+\delta )$ for some choice of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta > 0} .

To prove: If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(c)} exists, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(c) = 0}

Proof

Step no.	Assertion/construction	Facts used	Given data used	Previous steps used	Explanation
1	If the left hand derivative of $f$ at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} exists, then it is nonpositive, i.e., it is negative or zero.	Fact (2)	$f$ attains a (two-sided) local minimum at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c}		Since $f$ attains a two-sided local minimum at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} , it in particular attains a local minimum from the left. Thus, Fact (2) applies.
2	If the right hand derivative of $f$ at $c$ exists, then it is nonnegative, i.e., it is positive or zero.	Fact (4)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} attains a (two-sided) local minimum at $c$		Since $f$ attains a two-sided local minimum at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} , it in particular attains a local minimum from the right. Thus, Fact (4) applies.
3	If the (two-sided) derivative of $f$ exists at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} , it must be zero.			Steps (1), (2)	For the (two-sided) derivative to exist, both the left hand derivative and the right hand derivative must exist and they must be equal to each other and to the derivative. By Step (1), the derivative must be negative or zero. By Step (2), the derivative must be positive or zero. The only way both of these can be simultaneously true is if the derivative equals zero.

Note on strictness

Even if we consider the case of strict two-sided local maximum or strict two-sided local minimum, we can still have either of the two types of critical point: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(c) = 0} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(c)} not existing. The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(c) = 0} case continues to be possible because, for the one-sided versions, we can have the one-sided derivative equal to zero even assuming strict one-sided local extremum.