Question Number 86343 by jagoll last updated on 28/Mar/20
$$\mathrm{Dear}\:\mathrm{mr}\:\mathrm{w}.\:\mathrm{i}\:\mathrm{want}\: \\ $$$$\mathrm{discuss}\:\mathrm{for}\:\mathrm{equation} \\ $$$$\mathrm{find}\:\mathrm{minimum}\:\mathrm{and}\:\mathrm{maximum}\:\mathrm{value} \\ $$$$\mathrm{of}\:\mathrm{xy}\:+\mathrm{2}\:\mathrm{with}\:\mathrm{constraint} \\ $$$$\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}\:} =\:\mathrm{6}. \\ $$$$\mathrm{my}\:\mathrm{way}\:\left(\mathrm{short}\:\mathrm{cut}\right) \\ $$$$\Rightarrow\:\mathrm{x}^{\mathrm{2}} \:=\:\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{3} \\ $$$$\begin{cases}{\mathrm{max}\:=\:\left(\sqrt{\mathrm{3}}\right)^{\mathrm{2}} +\mathrm{2}\:=\:\mathrm{5}}\\{\mathrm{min}\:=\:−\left(\sqrt{\mathrm{3}}\right)^{\mathrm{2}} \:+\mathrm{1}\:=\:−\mathrm{1}}\end{cases} \\ $$$$\mathrm{it}\:\mathrm{correct}? \\ $$
Commented by mr W last updated on 28/Mar/20
$${this}\:{is}\:{correct},\:{because}\:{both}\:{in}\:{xy}+\mathrm{2} \\ $$$${and}\:{in}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{6}\:{there}\:{is}\:{the}\:{symmetry} \\ $$$${for}\:{x}\:{and}\:{y},\:{i}.{e}.\:{x}\:{and}\:{y}\:{are}\:{equally} \\ $$$${valued}. \\ $$
Commented by jagoll last updated on 28/Mar/20
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{mister} \\ $$