IMO 1970 LL FRA25

Suppose that f is a real function defined for 0 \leqx \leq1 having

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IMO 1970 LL FRA25

Origin: FRA

Problem

Suppose that f is a real function defined for 0 \leqx \leq1 having the first derivative f ′ for 0 \leqx \leq1 and the second derivative f ′′ for 0 < x < 1. Prove that if f(0) = f ′(0) = f ′(1) = f(1) −1 = 0, there exists a number 0 < y < 1 such that |f ′′(y)| \geq4.