Question Number 6179 by 314159 last updated on 17/Jun/16
$${Divide}\:\mathrm{10}\:{into}\:{two}\:{parts}\:{so}\:{that}\:{twice}\:{the} \\ $$$${square}\:{of}\:{the}\:{first}\:{part}\:{plus}\:{thrice}\:{the} \\ $$$${square}\:{of}\:{the}\:{other}\:{part}\:{is}\:{the}\:{least}. \\ $$
Commented by prakash jain last updated on 17/Jun/16
$${one}\:{part}\:{x},\:{second}\:{part}\:\left(\mathrm{10}−{x}\right) \\ $$$$\mathrm{Minimize} \\ $$$$\mathrm{2}{x}^{\mathrm{2}} +\mathrm{3}\left(\mathrm{10}−{x}\right)^{\mathrm{2}} \\ $$$$=\mathrm{2}{x}^{\mathrm{2}} +\mathrm{3}\left(\mathrm{100}−\mathrm{2}{x}+{x}^{\mathrm{2}} \right) \\ $$$$=\mathrm{5}{x}^{\mathrm{2}} −\mathrm{6}{x}+\mathrm{300} \\ $$$$=\mathrm{5}\left({x}^{\mathrm{2}} −\frac{\mathrm{6}}{\mathrm{5}}{x}+\mathrm{60}\right) \\ $$$$=\mathrm{5}\left({x}^{\mathrm{2}} −\frac{\mathrm{6}}{\mathrm{5}}{x}+\frac{\mathrm{9}}{\mathrm{25}}+\mathrm{60}−\frac{\mathrm{9}}{\mathrm{25}}\right) \\ $$$$=\mathrm{5}\left({x}^{\mathrm{2}} −\mathrm{2}\centerdot\left(\frac{\mathrm{3}}{\mathrm{5}}\right){x}+\left(\frac{\mathrm{3}}{\mathrm{5}}\right)^{\mathrm{2}} +\frac{\mathrm{1491}}{\mathrm{25}}\right) \\ $$$$=\mathrm{5}\left\{\left({x}−\frac{\mathrm{3}}{\mathrm{5}}\right)^{\mathrm{2}} +\frac{\mathrm{1491}}{\mathrm{25}}\right\} \\ $$$${minimum}\:{value}\:{when}\:{x}=\frac{\mathrm{3}}{\mathrm{5}} \\ $$$${since}\:\left({x}−\frac{\mathrm{3}}{\mathrm{5}}\right)^{\mathrm{2}} \geqslant\mathrm{0} \\ $$
Commented by Rasheed Soomro last updated on 18/Jun/16
$$\mathcal{N}{ice}!!! \\ $$$$ \\ $$