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?

Suppose $x\in [1/2,3)$ maximizes $f$ on $[1/2,3)$. But then, since $f$ is strictly increasing ($f'>0$), for any $y\in(x,3)$ we have $f(y)>f(x)$, a contradiction.
