Question Number 174472 by floor(10²Eta[1]) last updated on 01/Aug/22
![let f, g be continuous at [a,b], with f(x)≥0 at [a,b]. Prove that exists some θ∈[a,b] such that ∫_a ^b f(x)g(x)dx=g(θ)∫_a ^b f(x)dx](Q174472.png)
$$\mathrm{let}\:\mathrm{f},\:\mathrm{g}\:\mathrm{be}\:\mathrm{continuous}\:\mathrm{at}\:\left[\mathrm{a},\mathrm{b}\right],\:\mathrm{with}\:\mathrm{f}\left(\mathrm{x}\right)\geqslant\mathrm{0}\:\mathrm{at}\:\left[\mathrm{a},\mathrm{b}\right]. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{exists}\:\mathrm{some}\:\theta\in\left[\mathrm{a},\mathrm{b}\right]\:\mathrm{such}\:\mathrm{that} \\ $$$$\int_{\mathrm{a}} ^{\mathrm{b}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx}=\mathrm{g}\left(\theta\right)\int_{\mathrm{a}} ^{\mathrm{b}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ $$