The Chain Rule & Implicit Differentiation · Superchallenge problem

For each real $a$, let $V(a) = \displaystyle\max_{x\in\mathbb R}\,\big(ax - \cosh x\big)$. The maximiser $x^*(a)$ is a differentiable function of $a$ but you should not need a formula for $V$. Find $V''\!\left(\dfrac34\right)$.