JEE-Advanced 2015 (8/40)

Calculus Level 4

Let F(x)=xx2+π62cos2tdtF(x)=\displaystyle \int_{x}^{x^2+\frac{\pi}{6}} 2\cos^2t \, dt for all xRx\in \mathbb{R} and f:[0,12][0,)f:\left[0,\frac{1}{2} \right] \to [0,\infty) be a continuous function. For a[0,12]a \in \left[0,\frac{1}{2} \right], if F(a)+2F'(a)+2 is the area of the region bounded by x=0,y=0,y=f(x)x=0,y=0 ,y=f(x) and x=ax=a, then what is the f(0)f(0)?

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