Let $$F(x)=\displaystyle \int_{x}^{x^2+\frac{\pi}{6}} 2\cos^2t \, dt$$ for all $$x\in \mathbb{R}$$ and $$f:\left[0,\frac{1}{2} \right] \to [0,\infty)$$ be a continuous function. For $$a \in \left[0,\frac{1}{2} \right]$$, if $$F'(a)+2$$ is the area of the region bounded by $$x=0,y=0 ,y=f(x)$$ and $$x=a$$, then what is the $$f(0)$$?