JEE-Advanced 2015 (8/40)

Calculus Level 4

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 \(f(0)\)?

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