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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-




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 )
$${let}\:{f}:\left[\mathrm{0};\mathrm{1}\right]\rightarrow\mathbb{R},\:{prove}\:{that}\:\exists{x}_{\mathrm{0}} ,{x}_{\mathrm{1}} ,{x}_{\mathrm{2}} \in\left(\mathrm{0};\mathrm{1}\right) \\ $$$${such}\:{that}\:\:\frac{{f}\left({x}_{\mathrm{0}} \right)}{{x}_{\mathrm{0}} ^{\mathrm{2}} }+\frac{{f}\left({x}_{\mathrm{1}} \right)}{\mathrm{2}{x}_{\mathrm{1}} ^{\mathrm{2}} }=\mathrm{3}{f}\left({x}_{\mathrm{2}} \right) \\ $$

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