Question Number 202605 by hardmath last updated on 30/Dec/23
Commented by mr W last updated on 30/Dec/23
$${what}\:{do}\:{you}\:{mean}\:{when}\:{you}\:{write} \\ $$$$\underset{={b}} {{max}}\:\underset{={a}} {{min}}\:\left(…\right)\:? \\ $$
Commented by hardmath last updated on 30/Dec/23
$$\mathrm{dear}\:\mathrm{professor}, \\ $$$$ \\ $$In the given expression, the maximum value is for b, and the minimum value is for a
Commented by mr W last updated on 30/Dec/23
$${x}=\pm\mathrm{1} \\ $$
Answered by mr W last updated on 30/Dec/23
$${we}\:{replace}\:{a},\:{b},\:{x}\:{with}\:{x},\:{y},\:{k}. \\ $$$${f}\left({x},{y}\right)={x}^{\mathrm{2}} −\mathrm{2}{xy}−{y}^{\mathrm{2}} −\mathrm{2}{kx}+\mathrm{10}{ky} \\ $$$$\:\:\:=\left({x}−{y}−{k}\right)^{\mathrm{2}} −\mathrm{2}{y}^{\mathrm{2}} +\mathrm{8}{ky}−{k}^{\mathrm{2}} \\ $$$${f}_{{min}} {is}\:{when}\:{x}−{y}−{k}=\mathrm{0} \\ $$$${f}\left({x},{y}\right)=−\left({y}+{x}−\mathrm{5}{k}\right)^{\mathrm{2}} +\mathrm{2}{x}^{\mathrm{2}} −\mathrm{12}{kx}+\mathrm{25}{k}^{\mathrm{2}} \\ $$$${f}_{{max}} \:{is}\:{when}\:{x}+{y}−\mathrm{5}{k}=\mathrm{0} \\ $$$$\Rightarrow{x}=\mathrm{3}{k},\:{y}=\mathrm{2}{k} \\ $$$$\left(\mathrm{3}{k}\right)^{\mathrm{2}} −\mathrm{2}\left(\mathrm{3}{k}\right)\left(\mathrm{2}{k}\right)−\left(\mathrm{2}{k}\right)^{\mathrm{2}} −\mathrm{2}{k}\left(\mathrm{3}{k}\right)+\mathrm{10}{k}\left(\mathrm{2}{k}\right)=\mathrm{7} \\ $$$${k}^{\mathrm{2}} =\mathrm{1} \\ $$$$\Rightarrow{k}=\pm\mathrm{1} \\ $$
Commented by hardmath last updated on 30/Dec/23
$$\mathrm{cool}\:\mathrm{dear}\:\mathrm{professor}\:\mathrm{thank}\:\mathrm{you} \\ $$