Let $a$ and $b$ be two real numbers such that $0 < a < b$. Show that, for all $t > 0$ and all $x \in [a, b]$, $$t^{x} \leqslant \max\left(t^{a}, t^{b}\right) \leqslant t^{a} + t^{b}$$
Let $a$ and $b$ be two real numbers such that $0 < a < b$. Show that, for all $t > 0$ and all $x \in [a, b]$,
$$t^{x} \leqslant \max\left(t^{a}, t^{b}\right) \leqslant t^{a} + t^{b}$$