Showing there's no maximum in the given interval Let $f$ be function as $f(x) = x^2, x \in [\frac{1}{2}, 3).$ If we want to show there's no maximum in the given interval, is this the right way to do it:
Assume there's maximum $y \in (8, 9)$ for some $x \in [\frac{1}{2}, 3)$. Then show that there's some $y' \in (8, 9)$ for all $y \in (8, 9)$ such that $y' > y$ for some $x' \in [\frac{1}{2}, 3)$ so that we get a contradiction. Does that make sense?

Note that $x^2$ is increasing on this interval; thus its supremum is at the right end of the interval which isn't attained from the domain.
