Question Number 221899 by universe last updated on 12/Jun/25
![let a∈[0,1] find all continuous function f:[0,1]→[0,∞) such that ∫_0 ^1 f(x)dx = 1 , ∫_0 ^1 xf(x)dx = a and ∫_0 ^1 x^2 f(x)dx = a^2 how many such function are there ?](https://www.tinkutara.com/question/Q221899.png)
$$\:\:\:{let}\:{a}\in\left[\mathrm{0},\mathrm{1}\right]\:{find}\:{all}\:{continuous}\:{function} \\ $$$$\:\:\:\:\:{f}:\left[\mathrm{0},\mathrm{1}\right]\rightarrow\left[\mathrm{0},\infty\right)\:{such}\:{that}\:\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx}\:=\:\mathrm{1}\:\:, \\ $$$$\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} {xf}\left({x}\right){dx}\:=\:{a}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{\mathrm{2}} {f}\left({x}\right){dx}\:=\:{a}^{\mathrm{2}} \\ $$$$\:\:\:\:{how}\:{many}\:{such}\:{function}\:{are}\:{there}\:? \\ $$$$ \\ $$