JEE-Advanced 2015 (27/40)

Calculus Level 3

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

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