Suppose $X$ is a process given by -

$dX_t = db_t$ where $b_t$ is a standard Brownian motion with its filtration $(\mathcal{F}_t)$.

Suppose an agent earns a payoff given by

$V(x) = \mathbb{E} [\int_0^\infty e^{-\int_0^t r(X_s)ds} dt|X_0 =x] $

where $r(x) = \begin{cases} 3 & \text{ if } x \ge 0 \\ 7 & \text{ otherwise} \end{cases}$

I am interested in computing $V(x)$. In particular, I am interested in knowing if $V(x)$ is differentiable at $0$?

I suspect that the answer is no for differentiability. But I don't have a proof.

Thanks.