Chapter 12 - Functions of Several Veriables - 12.8 Maximum/Minimum Problems - 12.8 Exercises - Page 948: 24

$$\eqalign{ & {\text{Local maximum at }}\left( {1,1} \right) \cr & {\text{Saddle point at }}\left( {0,0} \right) \cr} $$

$$\eqalign{ & f\left( {x,y} \right) = xy{e^{ - x - y}} \cr & {\text{Calculate the first partial derivatives}} \cr & {f_x}\left( {x,y} \right) = y{e^{ - x - y}} - xy{e^{ - x - y}} \cr & {f_y}\left( {x,y} \right) = x{e^{ - x - y}} - xy{e^{ - x - y}} \cr & \cr & {\text{Calculate the second partial derivatives and }}{f_{xy}}\left( {x,y} \right) \cr & {f_{xx}}\left( {x,y} \right) = xy{e^{ - x - y}} - 2y{e^{ - x - y}} \cr & {f_{yy}}\left( {x,y} \right) = xy{e^{ - x - y}} - 2x{e^{ - x - y}} \cr & {f_{xy}}\left( {x,y} \right) = {e^{ - x - y}} - y{e^{ - x - y}} - x{e^{ - x - y}} + xy{e^{ - x - y}} \cr & {\text{Set the first partial derivatives to zero}}{\text{,}} \cr & {f_x}\left( {x,y} \right) = y{e^{ - x - y}} - xy{e^{ - x - y}} = 0\, \cr & {f_y}\left( {x,y} \right) = x{e^{ - x - y}} - xy{e^{ - x - y}} = 0\, \cr & y{e^{ - x - y}}\left( {1 - x} \right) = 0\,\,\,\left( {\bf{1}} \right) \cr & x{e^{ - x - y}}\left( {1 - y} \right) = 0\,\,\,\left( {\bf{2}} \right) \cr & {\text{From the equation }}\left( {\bf{1}} \right){\text{ we obtain}} \cr & x = 1,\,\,\,y = 0 \cr & {\text{From the equation }}\left( {\bf{2}} \right){\text{ we obtain}} \cr & x = 0,\,\,\,y = 1 \cr & \cr & {\text{The critical point are}} \cr & \left( {0,0} \right){\text{,}}\,\,\left( {1,1} \right) \cr & {\text{Evaluate the Second Derivative Test}} \cr & D\left( {x,y} \right) = {f_x}\left( {x,y} \right){f_y}\left( {x,y} \right) - {\left( {{f_{xy}}\left( {x,y} \right)} \right)^2} \cr & {\text{Evaluate }}D\left( {x,y} \right){\text{ at }}\left( {0,0} \right){\text{ and }}\left( {1,1} \right) \cr & D\left( {0,0} \right) = {f_x}\left( {0,0} \right){f_y}\left( {0,0} \right) - {\left( {{f_{xy}}\left( {0,0} \right)} \right)^2} \cr & D\left( {1,1} \right) = {f_x}\left( {1,1} \right){f_y}\left( {1,1} \right) - {\left( {{f_{xy}}\left( {1,1} \right)} \right)^2} = 32 \cr & {\text{Using a calculator we obtain}} \cr & D\left( {0,0} \right) = - 1 \cr & D\left( {1,1} \right) \approx 0.0183 \cr & \cr & {\text{Evaluate }}{f_{xx}}\left( {x,y} \right){\text{ at }}\left( {1,1} \right) \cr & {f_{xx}}\left( {1,1} \right) \approx - 0.135 \cr & \cr & {\text{Using the second derivative test}}{\text{, we conclude that}} \cr & D\left( {0,0} \right) < 0,\,\,\, \Rightarrow {\text{saddle point at }}\left( {0,0} \right) \cr & D\left( {1,1} \right) > 0,\,\,\,\,{f_{xx}}\left( {1,1} \right) < 0 \Rightarrow {\text{Local maximu at }}\left( {1,1} \right) \cr} $$