Question Number 207315 by galva2000 last updated on 11/May/24
$${if}\:{ab}+{ac}+{bc}=\mathrm{2}\: \\ $$$${calculate}\:{minimum}\:{of}\:\mathrm{10}{a}^{\mathrm{2}} +\mathrm{10}{b}^{\mathrm{2}} +{c}^{\mathrm{2}} \\ $$
Answered by Berbere last updated on 11/May/24
$${S}=\mathrm{10}{a}^{\mathrm{2}} +\mathrm{10}{b}^{\mathrm{2}} +{c}=\mathrm{8}{a}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}{c}^{\mathrm{2}} +\mathrm{2}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)+\mathrm{8}{b}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}{c}^{\mathrm{2}} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} \geqslant\mathrm{2}{xy} \\ $$$${S}\geqslant\mathrm{4}{ac}+\mathrm{4}{ab}+\mathrm{4}{bc}=\mathrm{4}\left({ab}+{bc}+{ac}\right)=\mathrm{8} \\ $$$${Min}=\mathrm{8}\: \\ $$$$ \\ $$$$ \\ $$$$ \\ $$