Chapter 12 - Functions of Several Veriables - 12.8 Maximum/Minimum Problems - 12.8 Exercises - Page 948: 23

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

$$\eqalign{ & f\left( {x,y} \right) = {x^4} + 2{y^2} - 4xy \cr & {\text{Calculate the first partial derivatives}} \cr & {f_x}\left( {x,y} \right) = 4{x^3} - 4y \cr & {f_y}\left( {x,y} \right) = 4y - 4x \cr & {\text{Calculate the second partial derivatives and }}{f_{xy}}\left( {x,y} \right) \cr & {f_{xx}}\left( {x,y} \right) = 12{x^2} \cr & {f_{yy}}\left( {x,y} \right) = 4 \cr & {f_{xy}}\left( {x,y} \right) = - 4 \cr & {\text{Set the first partial derivatives to zero}}{\text{,}} \cr & {f_x}\left( {x,y} \right) = 4{x^3} - 4y = 0\,\,\left( {\bf{1}} \right) \cr & {f_y}\left( {x,y} \right) = 4y - 4x = 0\,\,\,\,\left( {\bf{2}} \right) \cr & {\text{From }}\left( {\bf{2}} \right) \cr & y = x \cr & \left( {\bf{2}} \right){\text{ into }}\left( {\bf{1}} \right) \cr & 4{x^3} - 4x = 0 \cr & 4x\left( {{x^2} - 1} \right) = 0 \cr & x = 0,\,\,\,x = - 1,\,\,\,x = 1 \cr & \cr & {\text{The critical point are}} \cr & \left( { - 1, - 1} \right),\,\left( {0,0} \right){\text{ and }}\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( { - 1, - 1} \right),\,\left( {0,0} \right){\text{ and }}\left( {1,1} \right) \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 & 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} = - 16 \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 & \cr & {\text{Evaluate }}{f_{xx}}\left( {x,y} \right){\text{ at }}\left( { - 1, - 1} \right),\,\left( {0,0} \right){\text{ and }}\left( {1,1} \right) \cr & {f_{xx}}\left( { - 1, - 1} \right) = 12{\left( { - 1} \right)^2} = 12 \cr & {f_{xx}}\left( {0,0} \right) = 12{\left( 0 \right)^2} = 0 \cr & {f_{xx}}\left( {1,1} \right) = 12{\left( 1 \right)^2} = 12 \cr & \cr & {\text{Using the second derivative test}}{\text{, we conclude that}} \cr & D\left( { - 1, - 1} \right) > 0,\,\,\,\,{f_{xx}}\left( { - 1, - 1} \right) > 0 \Rightarrow {\text{Local minimum at }}\left( { - 1, - 1} \right) \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 minimum at }}\left( {1,1} \right) \cr} $$