# Find-the-local-extreme-values-of-the-function-f-x-y-xy-x-2-y-2-2x-2y-4-

Question Number 69162 by Learner-123 last updated on 20/Sep/19
$${Find}\:{the}\:{local}\:{extreme}\:{values}\:{of} \\$$$${the}\:{function}\:: \\$$$${f}\left({x},{y}\right)=\:{xy}−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} −\mathrm{2}{x}−\mathrm{2}{y}+\mathrm{4}. \\$$
Answered by MJS last updated on 20/Sep/19
$$\frac{{d}}{{dx}}\left[−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} +{xy}−\mathrm{2}{x}−\mathrm{2}{y}+\mathrm{4}\right]=−\mathrm{2}{x}+{y}−\mathrm{2} \\$$$$−\mathrm{2}{x}+{y}−\mathrm{2}=\mathrm{0}\:\Rightarrow\:{x}=\frac{{y}−\mathrm{2}}{\mathrm{2}} \\$$$${f}\left(\frac{{y}−\mathrm{2}}{\mathrm{2}},{y}\right)=−\frac{\mathrm{3}}{\mathrm{4}}{y}^{\mathrm{2}} −\mathrm{3}{y}+\mathrm{5} \\$$$$\frac{{d}}{{dy}}\left[−\frac{\mathrm{3}}{\mathrm{4}}{y}^{\mathrm{2}} −\mathrm{3}{y}+\mathrm{5}\right]=−\frac{\mathrm{3}}{\mathrm{2}}{y}−\mathrm{3} \\$$$$−\frac{\mathrm{3}}{\mathrm{2}}{y}−\mathrm{3}=\mathrm{0}\:\Rightarrow\:{y}=−\mathrm{2}\:\Rightarrow\:{x}=−\mathrm{2} \\$$$$\mathrm{the}\:\mathrm{2}^{\mathrm{nd}} \:\mathrm{derivates}\:\mathrm{are}\:<\mathrm{0}\:\Rightarrow\:\mathrm{maximum}\:\mathrm{at} \\$$$$\begin{pmatrix}{{x}}\\{{y}}\\{{f}\left({x},\:{y}\right)}\end{pmatrix}\:=\begin{pmatrix}{−\mathrm{2}}\\{−\mathrm{2}}\\{\mathrm{8}}\end{pmatrix} \\$$
Commented by Learner-123 last updated on 23/Sep/19
$${thanks}\:{sir}. \\$$