Question Number 96185 by bemath last updated on 30/May/20
$$\mathrm{what}\:\mathrm{are}\:\mathrm{critical}\:\mathrm{points}\:\mathrm{of}\:\mathrm{this} \\ $$$$\mathrm{function}\:\mathrm{z}\:=\:\mathrm{xy}+\mathrm{5xy}^{\mathrm{2}} +\mathrm{10y} \\ $$
Answered by john santu last updated on 30/May/20
$$\left(\mathrm{1}\right)\:\frac{\partial\mathrm{z}}{\partial{x}}\:=\:{y}+\mathrm{5}{y}^{\mathrm{2}} \:=\:\mathrm{0}\: \\ $$$${y}\left(\mathrm{1}+\mathrm{5}{y}\right)\:=\:\mathrm{0}\:\begin{cases}{{y}\:=\:\mathrm{0}}\\{{y}=−\frac{\mathrm{1}}{\mathrm{5}}}\end{cases} \\ $$$$\left(\mathrm{2}\right)\:\frac{\partial\mathrm{z}}{\partial\mathrm{y}}\:=\:{x}+\mathrm{10}{x}\mathrm{y}+\mathrm{10}=\mathrm{0} \\ $$$$\Rightarrow{x}\left(\mathrm{1}+\mathrm{10y}\right)=−\mathrm{10}. \\ $$$$\mathrm{if}\:\mathrm{y}\:=\:\mathrm{0}\:\Rightarrow\:{x}\:=\:−\mathrm{10} \\ $$$$\mathrm{if}\:\mathrm{y}\:=\:−\frac{\mathrm{1}}{\mathrm{5}}\Rightarrow{x}\:=\:\mathrm{10} \\ $$$${critical}\:{point}\:{are}\:\left(−\mathrm{10},\mathrm{0}\right)\:;\: \\ $$$$\left(\mathrm{10},−\frac{\mathrm{1}}{\mathrm{5}}\right)\:. \\ $$$$ \\ $$