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easy gronwall type inequality

Source: 2016 South Korea USCM P4

August 16, 2020
inequalitiesreal analysismean value theoremcollege contestsfunction

Problem Statement

Suppose a continuous function f:[−π4,π4]→[−1,1]f:[-\frac{\pi}{4},\frac{\pi}{4}]\to[-1,1] and differentiable on (−π4,π4)(-\frac{\pi}{4},\frac{\pi}{4}). Then, there exists a point x0∈(−π4,π4)x_0\in (-\frac{\pi}{4},\frac{\pi}{4}) such that ∣f′(x0)∣≤1+f(x0)2|f'(x_0)|\leq 1+f(x_0)^2
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