I want to rigorously agrue that for $x,y \in \mathcal{C}\big([a,b], [0,\infty)\big)$ holds
$$ \max_{t \in [a,b]} \{x(t)\cdot y(t)\} \le \max_{t \in [a,b]} x (t)\cdot \max_{t \in [a,b]} y(t).$$
While it seems fairly obvious to me that this is true, due to the positivity of the functions $x,y$, how can this be proven or related to known characteristics of the $\max$ from calculus?
Idea: Let $$\alpha := \max_{t \in [a,b]} \{x(t)\cdot y(t)\}, \ \beta := \max_{t \in [a,b]} x (t), \ \gamma := \max_{t \in [a,b]} y(t),$$ then for $\alpha = \beta \cdot \gamma$ there is nothing to show. Claim: If $\alpha \neq \beta \cdot \gamma$, then $\alpha < \beta \cdot \gamma$. Suppose $\alpha > \beta \cdot \gamma.$ W.l.o.g assume $t_0 \neq t_1 \neq t_2$ and $$ x(t_0)\cdot y(t_0) > x(t_1)\cdot y(t_2).$$
No need for case distinction. Preliminary remark: The maxima are attained because we have continuous functions on a compact. So for some $\tau \in [a,b]$, we have $\alpha=x(\tau)y(\tau)$. Then from $0\le x(\tau)\le\beta$ and $0\le y(\tau)\le \gamma$, we obtain $(0\le)\,x(\tau)y(\tau)\le\beta\gamma$, whence the claim.