Question on the existence of a maximum for a continuous function

Question

Which definition or theorem should I use to prove this? It is obvious if I try to draw some graphs, but I am confused to prove it formally

Answer 1

You can use the extreme value theorem, with a little extra work. Because $\lim_{x \to \infty} f(x) = 0$, you know that $f(x) \le 1$ eventually. That is, you should be able to pick some $M$ such that $$x > M \implies f(x) \le 1.$$ Then, you can consider $f$ restricted to $[0, M]$. Since it's continuous on this compact interval, it attains its maximum at some point $x_0 \in [0, M]$. Note that $f(x_0) \ge f(0) = 1$.

I claim that this is a maximum over all of $[0, \infty)$. If we have $x \in [0, \infty)$, then either $x > M$ or $x \le M$. If $x \le M$, then $x \in [0, M]$, so $f(x) \le f(x_0)$ by construction. If $x > M$, then $$f(x) \le 1 = f(0) \le f(x_0).$$ Either way, $f(x) \le f(x_0)$, so $x_0$ is the global maximum of the function.