Question Number 140392 by mathdanisur last updated on 07/May/21
![let f:[0;1]→R, prove that ∃x_0 ,x_1 ,x_2 ∈(0;1) such that ((f(x_0 ))/x_0 ^2 )+((f(x_1 ))/(2x_1 ^2 ))=3f(x_2 )](Q140392.png)
$${let}\:{f}:\left[\mathrm{0};\mathrm{1}\right]\rightarrow\mathbb{R},\:{prove}\:{that}\:\exists{x}_{\mathrm{0}} ,{x}_{\mathrm{1}} ,{x}_{\mathrm{2}} \in\left(\mathrm{0};\mathrm{1}\right) \\ $$$${such}\:{that}\:\:\frac{{f}\left({x}_{\mathrm{0}} \right)}{{x}_{\mathrm{0}} ^{\mathrm{2}} }+\frac{{f}\left({x}_{\mathrm{1}} \right)}{\mathrm{2}{x}_{\mathrm{1}} ^{\mathrm{2}} }=\mathrm{3}{f}\left({x}_{\mathrm{2}} \right) \\ $$