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

  • 1
    $\begingroup$ Yes that is true. $\endgroup$ – max_zorn May 15 '18 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 May 31 '18 at 21:15

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.