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Prove this integral trigonometric inequality

Source: 2019 Jozsef Wildt International Math Competition-W. 16

May 18, 2020
integrationtrigonometryinequalitiescalculus

Problem Statement

If f:[a,b]→(0,∞)f : [a, b] \to (0,\infty); 0<a≤b0 < a \leq b; ff derivable; f′f' continuous then:∫abf′(x)f(x)f3(x)+1≤tan⁡−1(f(b)−f(a)1+f(a)f(b))\int \limits_{a}^{b}\frac{f'(x)\sqrt{f(x)}}{f^3(x) + 1}\leq \tan^{-1}\left(\frac{f(b)-f(a)}{1 + f(a)f(b)}\right)
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