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Question Number 174472 by floor(10²Eta[1]) last updated on 01/Aug/22

$$\mathrm{let}\mathrm{f},\mathrm{g}\mathrm{be}\mathrm{continuous}\mathrm{at}[\mathrm{a},\mathrm{b}],\mathrm{with}\mathrm{f}\left(\mathrm{x}\right)⩾0\mathrm{at}[\mathrm{a},\mathrm{b}].$$$$\mathrm{Prove}\mathrm{that}\mathrm{exists}\mathrm{some}\theta \in [\mathrm{a},\mathrm{b}]\mathrm{such}\mathrm{that}$$$${\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx}=\mathrm{g}\left(\theta \right){\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$$

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