Exsitence of extrema and boundedness of a function Let $f:[0, \infty) \longrightarrow \mathbb R$ be a continuous function which converges for $t \rightarrow +\infty$.

What can you say about existence of extrema and boundedness of the function f?
If $\Vert f \Vert = \sup_{x \in [0,\infty)} \frac{|f(x)|}{|x|} &lt; \infty$ Holds it is bounded. If the image of $f$ is a compact set it attains its maximum and minimum
If $\Vert f \Vert = \sup_{x \in [0,\infty)} \frac{|f(x)|}{|x|} &lt; \infty$ Holds it is bounded. If the image of $f$ is a compact set it attains its maximum and minimum