Maximum

[Papapetros Vaggelis]

Let \(\displaystyle{f:\left[0,+\infty\right)\longrightarrow \mathbb{R}}\) be a continuous function such that

\(\displaystyle{f(0)=2}\) and \(\displaystyle{\lim_{x\to +\infty}f(x)=0}\).

Prove that the function \(\displaystyle{f}\) has a maximum value at \(\displaystyle{\left[0,+\infty\right)}\) .

[Riemann]

Hello Vaggelis. Here is a solution:

The second condition tells us that there exists an $M>0$ such that for every $ x> M$ to hold $f(x)<2$. Since $f$ is continuous on $[0, M]$ a maximum is attained and let that be at $f(x_0)$. Then:

$$f(x_0) \geq f(x) \;\; \forall x \in [0, M] \quad \quad \text{and} \quad \quad f(x_0) \geq f(0)= 2 > f(x) \; \; \forall x \in [M, +\infty)$$

Hence $f(x_0)$ is maximum of $f$.