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Does a local maximum mean the function(excluding the constant function) is increasing to the left and decreasing to the right in some nearby interval of the critical point ?

I am firmly sure this it is true as it is so intuitive,but some one says this is not true, so anyone can give a convincing explanation here ?

What a local maximum means I learned is just that no nearby output value is greater than that local maximum value.

P.S.thank you everyone, but I must declare I only have some basic knowledge of derivatives ,I started to learn calculus a few days ago .

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No. Take for example $$f(x)=\left\{ \begin{array}{cl} -x&\text{ if }x<0\\ 1&\text{ if }x=0\\ x&\text{ if }x>0 \end{array} \right.$$

This function has a maximum at $x=0$, but it is decreasing in $(-\infty,0)$ and increasing in $(0,\infty)$.

Not even the continuity at the maximum suffices to guarantee this: $$f(x)=\left\{ \begin{array}{cl} x+\frac x3\sin\left(\frac 1x\right)&\text{ if }x<0\\ 0&\text{ if }x=0\\ -x+\frac x3\sin\left(\frac 1x\right)&\text{ if }x>0 \end{array} \right.$$

This function has a maximum at $x=0$ and it is continuous at $x=0$, but there's no interval $(-\infty, 0]$ in which $f$ is increasing, or even monotonic. The same for $[0,\infty)$. To illustrate this, compute for example $f'(\frac1{300\pi})$ and $f'(\frac1{301\pi})$.

If the function is derivable and the derivative is continuous at the maximum, your statement is true.

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  • $\begingroup$ (1) The first function has a local maximum at x=0. (2) The second example is indicating that a point of local maximum does not have to be "increasing to the left and decreasing to the right". $\endgroup$ – Yuan Gao Nov 7 '14 at 5:10
  • $\begingroup$ “If the function is derivable and the derivative is continuous at the maximum, your statement is true.” how to prove it? $\endgroup$ – iMath Dec 10 '14 at 14:36
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It depends on your definitions of local maximum and increasing. If (for example) you take the definitions from Stewart's Calculus: Concepts and Contexts then a constant function has a local maximum at every point but is nowhere increasing or decreasing.

For reference, Stewart's definitions:

The number $f(c)$ is a local maximum of $f$ if $f(c) \geq f(x)$ when $x$ is near $c$.

A function is increasing on an interval $I$ if $f(x_{1}) < f(x_{2})$ whenever $x_{1} < x_{2}$.

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  • $\begingroup$ sorry for my vague question before,I've updated my question, I mean that just in some nearby interval of the critical point. $\endgroup$ – iMath Nov 4 '14 at 3:35
  • $\begingroup$ @iMath If that is your definition then a constant function, e.g. $f(x) = 5$ has a local maximum at every point (since there are no nearby values greater than $5$) but it's not increasing or decreasing (depending on your definition of increasing and decreasing). $\endgroup$ – in_mathematica_we_trust Nov 4 '14 at 6:58
  • $\begingroup$ N.B. excluding the constant function here $\endgroup$ – iMath Nov 5 '14 at 5:22

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