Let's have some generalization

Calculus Level 5

\[\begin{align} f(f(x))&=x \quad \forall x \in [0,1] \\ f(0)&=1\\ \exists x \in (0,1) \to f(x) &\neq 1-x\\ \exists x \in (0,1) \to f(x) &\neq \frac{1-x}{1+x} \end{align} \]

Let \(f\) be a differentiable function defined on \([0,1]\) and satisfies the conditions above .

Find the value of the following integral, up to three decimal places. \[10^5 \int_0^1 (x-f(x))^{2016} \,dx\]


Source: ISI entrance examination.
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