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I'm studying Single Variable Calculus (E7) by James Stewart. In Chapter 4.1, the book does not have a clear statement about the relationship between absolute max/min and local max/min. This is my proposition:

Function $f$ is continuous on a closed interval $[a, b]$. If $c\in(a, b)$ and $f(c)$ is the absolute max/min, then $f(c)$ must also be a local max/min for any open interval containing c and within domain $[a, b]$.

Is this always true?

Related question:

Closed Interval Method

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    $\begingroup$ Yes that is true. $\endgroup$
    – max_zorn
    Commented May 15, 2018 at 7:00
  • $\begingroup$ @user2923419 Please recall that if the OP is solved you can evaluate to accept an answer among the given, more details here meta.stackexchange.com/questions/5234/… $\endgroup$
    – user
    Commented May 31, 2018 at 21:15

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Yes it is always true for f continuous that if the absolute max/min value occurs for such $c$ it must be also a local max/min value for any open interval containing $c$ and within $[a,b]$.

It is not true in general for $c=a,b$. Consider for example $y=x$ in $[-1,1]$.

The statement can be easily proved by the definition of local max/min.

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