MathDB
f(0)=1, f(x) \ge 0 \ge f'(x), f"(x)\le f'(x) for x\ge 0

Source: ISI BS 2011 P4

March 31, 2013
functionintegrationinequalitiescalculuscalculus computations

Problem Statement

Let ff be a twice differentiable function on the open interval (−1,1)(-1,1) such that f(0)=1f(0)=1. Suppose ff also satisfies f(x)≥0,f′(x)≤0f(x) \ge 0, f'(x) \le 0 and f′′(x)≤f(x)f''(x) \le f(x), for all x≥0x\ge 0. Show that f′(0)≥−2f'(0) \ge -\sqrt2.
Back to ProblemsView on AoPS