Chapter 13 - Functions of Several Variables - Review Exercises - Page 962: 62

$${\text{Saddle point at }}\left( {8, - 8,0} \right)$$

$$\eqalign{ & f\left( {x,y} \right) = {x^2} - {y^2} - 16x - 16y \cr & {\text{Calculate the first partial derivatives of }}f\left( {x,y} \right) \cr & {f_x}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {{x^2} - {y^2} - 16x - 16y} \right] \cr & {f_x}\left( {x,y} \right) = 2x - 16 \cr & and \cr & {f_y}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {{x^2} - {y^2} - 16x - 16y} \right] \cr & {f_y}\left( {x,y} \right) = - 2y - 16 \cr & {\text{Setting both first partial derivatives equal to zero, we have}} \cr & {f_x}\left( {x,y} \right) = 0,{\text{ }}{f_y}\left( {x,y} \right) = 0 \cr & 2x - 16 = 0,{\text{ }} - 2y - 16 = 0 \cr & x = 8,{\text{ }}y = - 8 \cr & {\text{The critical point is }}\left( {8, - 8} \right) \cr & {\text{Find the second partial derivatives of }}f\left( {x,y} \right){\text{ and }}{f_{xy}}\left( {x,y} \right) \cr & {f_{xx}}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {2x - 16} \right] = 2 \cr & {f_{yy}}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ { - 2y - 16} \right] = - 2 \cr & {f_{xy}}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {2x - 16} \right] = 0 \cr & {\text{By the second partials test}} \cr & d\left( {x,y} \right) = {f_{xx}}\left( {a,b} \right){f_{yy}}\left( {a,b} \right) - {\left[ {{f_{xy}}\left( {x,y} \right)} \right]^2} \cr & {\text{Evaluate }}d\left( {x,y} \right){\text{ at the critical point }}\left( {8, - 8} \right) \cr & d\left( {8, - 8} \right) = {f_{xx}}\left( {8, - 8} \right){f_{yy}}\left( {8, - 8} \right) - {\left[ {{f_{xy}}\left( {8, - 8} \right)} \right]^2} \cr & d\left( {8, - 8} \right) = \left( 2 \right)\left( { - 2} \right) - {\left[ 0 \right]^2} \cr & d\left( {8, - 8} \right) = - 4 \cr & d < 0,{\text{ then}} \cr & f\left( {x,y} \right){\text{ has a saddle point at }}\left( {8, - 8,f\left( {8, - 8} \right)} \right) \cr & f\left( {x,y} \right) = {x^2} - {y^2} - 16x - 16y \cr & f\left( {8, - 8} \right) = {\left( 8 \right)^2} - {\left( { - 8} \right)^2} - 16\left( 8 \right) - 16\left( { - 8} \right) \cr & f\left( {8, - 8} \right) = 0 \cr & {\text{Saddle point at }}\left( {8, - 8,0} \right) \cr} $$