JEE-Advanced 2015 (27/40)

Calculus Level 3

Let f:RRf:\mathbb R \to \mathbb R be a continuous odd function, which vanishes exactly at one point and f(1)=12f(1)=\frac{1}{2}. Suppose that F(x)=1xf(t)dtF(x)=\displaystyle \int_{-1}^x f(t)dt for all x[1,2]x \in [-1,2] and G(x)=1xtf(f(t))dtG(x)=\displaystyle \int_{-1}^x t|f(f(t))|dt for all x[1,2]x \in [-1,2]. Find the value of f(12)f \left(\frac{1}{2} \right) if limx1F(x)G(x)=114\displaystyle \lim_{x \to 1} \frac{F(x)}{G(x)}=\frac{1}{14}

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