If $f(x):[0,\infty)\to\mathbb{R}$ is continuous with $\lim_{x\to\infty}f(x)=0$, then there need not exist a maximum in $[0,\infty)$

Question

If $f(x):[0,\infty)\to\mathbb{R}$ is continuous with $\lim_{x\to\infty}f(x)=0$, then does there exist a maximum in $[0,\infty)$ of $f$?

I think it should, but not sure how to prove it. Now, if there does not exist a maximum, it should either diverge at $0$ (by continuity) or oscillate infinitely at $0$. But, it is well defined on the closed interval $[0,\infty)$. Any examples of this kind? Thanks beforehand.

Answer 1

Let consider the case

• $f(x)=-(1+x)e^{-x}$

Note that to guarantee a maximum by EVT we need to set $f(0)=0$.

Answer 2

Just take $f(x)=-\frac1{x+1}$. It has no maximum.