Let $F(x)=\int_x^{x^2+\frac{\pi}{6}} 2 \cos ^2 t d t$ for all $x \in R$ and $f:\left[0, \frac{1}{2}\right] \rightarrow[0, \infty)$ be a continuous function. For $a \in\left[0, \frac{1}{2}\right]$, if $F^{\prime}(a)+2$ is the area of the region bounded by $x=0, y=0, y=f(x)$ and $x=a$, then $f(0)$ is