MathDB
Integral inequality

Source: Romanian District Olympiad 2023 12.1

March 11, 2023
real analysisIntegralinequalities

Problem Statement

Let f:[−π/2,π/2]→Rf:[-\pi/2,\pi/2]\to\mathbb{R} be a twice differentiable function which satisfies (f′′(x)−f(x))⋅tan⁡(x)+2f′(x)⩾1,\left(f''(x)-f(x)\right)\cdot\tan(x)+2f'(x)\geqslant 1,for all x∈(−π/2,π/2)x\in(-\pi/2,\pi/2). Prove that ∫−π/2π/2f(x)⋅sin⁡(x) dx⩾π−2.\int_{-\pi/2}^{\pi/2}f(x)\cdot \sin(x) \ dx\geqslant \pi-2.
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