Question 159
Let $n \in \mathbb{N}$, $n \geq 2$, and let $a_{1}, a_{2}, \ldots, a_{n} \in (1, \infty)$. Prove that the function $f:[0, \infty) \rightarrow \mathbb{R}$ defined by
$$
f(x)=\left(a_{1} a_{2} \ldots a_{n}\right)^{x}-a_{1}^{x}-a_{2}^{x}-\ldots-a_{n}^{x}
$$
for every $x \in [0, \infty)$ is strictly increasing.
$$
f(x)=\left(a_{1} a_{2} \ldots a_{n}\right)^{x}-a_{1}^{x}-a_{2}^{x}-\ldots-a_{n}^{x}
$$
for every $x \in [0, \infty)$ is strictly increasing.