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Question Number 107283 by mathmax by abdo last updated on 09/Aug/20

$$\mathrm{f}\mathrm{integrable}\mathrm{continue}\mathrm{on}[\mathrm{a},\mathrm{b}]\mathrm{let}\mathrm{m}=\inf \mathrm{f}\left(\mathrm{x}\right)\mathrm{and}\mathrm{M}=\sup \mathrm{f}\left(\mathrm{x}\right)$$$$(\mathrm{x}\in [\mathrm{a},\mathrm{b}]\mathrm{prove}\mathrm{that}{(\mathrm{b}-\mathrm{a})}^2⩽{\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\times {\int }_{\mathrm{a}}^{\mathrm{b}}\frac{\mathrm{dx}}{\mathrm{f}\left(\mathrm{x}\right)}⩽\frac{{(\mathrm{b}-\mathrm{a})}^2}{4}\frac{{(\mathrm{m}+\mathrm{M})}^2}{\mathrm{mM}}$$

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