Chapter 4, Problem 1SYU

To determine

To indicate:

Whether the statement is true or false.

Explanation of Solution

Given information:

The statement “The function f has a local maximum at p if $f\left(p\right)\le f\left(x\right)$ for points x near p.

A local maximum point on a function is a point $\left(p,f\left(p\right)\right)$ on the graph of the function whose y coordinate is larger than all other $y$ coordinates on the graph at points "close to'' $\left(p,f\left(p\right)\right)$. More precisely, $\left(p,f\left(p\right)\right)$ is a local maximum if there is an interval $\left(a,b\right)$ with $a<p<b$ and $f\left(p\right)\ge f\left(x\right)$ for every $x$ in $\left(a,b\right)$.

Similarly, the critical point $\left(p,f\left(p\right)\right)$is a local minimum point if it has locally the smallest $y$ coordinate. Again being more precise: $\left(p,f\left(p\right)\right)$ is a local minimum if there is an interval $\left(a,b\right)$ with $a<p<b$ and $f\left(p\right)\le f\left(x\right)$ for every $x$ in $\left(a,b\right)$. A local extremum is either a local minimum or a local maximum.