Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that its derivative $f^{\prime}$ is continuous and $f(\pi)=-6$. If $F:[0, \pi] \rightarrow \mathbb{R}$ is defined by $F(x)=\int_{0}^{x} f(t) d t$, and if \[ \int_{0}^{\pi}\left(f^{\prime}(x)+F(x)\right) \cos x d x=2, \] then what is the value of $f(0)$?

JEE · Math · previous-year question