Let $f:[0, +\infty) \rightarrow \mathbb{R}$ be a continuous function such that $f(0)=\ell+1 $ and $\displaystyle \lim_{x\rightarrow +\infty} f(x)=\ell$. Prove that $f$ has maximum.
Solution
Since $\displaystyle \lim_{x\rightarrow +\infty} f(x)=\ell$ we get that for all $x >M$ holds $f(x)<\ell+1/2$. On the other hand, $f$ is continuous on $[0, M]$ , hence a maximum is attained and let us call it $f(x_0)$. Then:
$$f\left ( x_0 \right )\geq f(x), \;\; \forall x \in [0, M] \;\;\; {\rm and} \;\;\; f(x_0)\geq f(0)=\ell +1 \geq f(x) \;\; \forall x \in [M, +\infty)$$
Hence $f(x_0)$ is the maximum of $f$.
Solution
Since $\displaystyle \lim_{x\rightarrow +\infty} f(x)=\ell$ we get that for all $x >M$ holds $f(x)<\ell+1/2$. On the other hand, $f$ is continuous on $[0, M]$ , hence a maximum is attained and let us call it $f(x_0)$. Then:
$$f\left ( x_0 \right )\geq f(x), \;\; \forall x \in [0, M] \;\;\; {\rm and} \;\;\; f(x_0)\geq f(0)=\ell +1 \geq f(x) \;\; \forall x \in [M, +\infty)$$
Hence $f(x_0)$ is the maximum of $f$.
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