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Question Number 163209 by HongKing last updated on 04/Jan/22

$$\mathrm{f}:\mathrm{I}\to (0;\mathrm{\infty });\mathrm{I}\subset \mathbb{R}$$$$\mathrm{f}-\mathrm{twice}\mathrm{derivable};\mathrm{f}{}^{{}^{\prime }};\mathrm{f}{}^{{}^{″}}-\mathrm{continuous}$$$$\mathrm{f}{}^{{}^{″}}\left(\mathrm{x}\right)\mathrm{f}\left(\mathrm{x}\right)⩾{\left(\mathrm{f}{}^{{}^{\prime }}\left(\mathrm{x}\right)\right)}^2;\mathrm{\forall }\mathrm{x}\in \mathrm{I}$$$$\mathrm{then}\mathrm{prove}\mathrm{that}:$$$$2\mathrm{f}\left(\frac{\mathrm{x}+\mathrm{y}}{2}\right)⩽\mathrm{f}\left(\mathrm{x}\right)+\mathrm{f}\left(\mathrm{y}\right);\mathrm{\forall }\mathrm{x};\mathrm{y}\in \mathrm{I}$$

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