Let $(X_1, \ldots, X_n)$ be discrete real random variables that are mutually independent such that, for all $k \in \{1, \ldots, n\}$,
$$P[X_k = 1] = P[X_k = -1] = \frac{1}{2}$$
For each $\lambda \geqslant 0$, we set
$$m(\lambda) = \frac{E[X_1 \exp(\lambda X_1)]}{E[\exp(\lambda X_1)]}$$
Show that the function $m$ is strictly increasing on $\mathbb{R}_{\geqslant 0}$, and that for all $t \in [0,1]$, there exists a unique $\lambda \geqslant 0$ such that $m(\lambda) = t$.