Chapter 13 - Partial Derivatives - Chapter 13 Review Exercises - Page 998: 33

$${\text{relative mimimun at }}\left( {15, - 8} \right)$$

$$\eqalign{ & f\left( {x,y} \right) = {x^2} + 3xy + 3{y^2} - 6x + 3y \cr & \,\,\,\,\,{f_x}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {{x^2} + 3xy + 3{y^2} - 6x + 3y} \right] \cr & \,\,\,\,\,{f_x}\left( {x,y} \right) = 2x + 3y - 6 \cr & \,\,\,\,\,{f_{xx}}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {2x + 3y - 6} \right] \cr & \,\,\,\,\,{f_{xx}}\left( {x,y} \right) = 2 \cr & and \cr & \,\,\,\,\,{f_y}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {{x^2} + 3xy + 3{y^2} - 6x + 3y} \right] \cr & \,\,\,\,\,{f_y}\left( {x,y} \right) = 3x + 6y + 3 \cr & \,\,\,\,\,{f_{yy}}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {3x + 6y + 3} \right] \cr & \,\,\,\,\,{f_{yy}}\left( {x,y} \right) = 6 \cr & {\text{Calculate the mixed partial derivative }}{f_{xy}}\left( {x,y} \right) \cr & \,\,\,\,\,{f_{xy}}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {2x + 3y - 6} \right] \cr & \,\,\,\,\,{f_{xy}}\left( {x,y} \right) = 3 \cr & {\text{Setting the first partial derivatives to 0}} \cr & {f_x}\left( {x,y} \right) = 2x + 3y - 6 = 0{\text{ and }}{f_x}\left( {x,y} \right) = 3x + 6y + 3 = 0 \cr & 2x + 3y = 6\,\,\left( 1 \right),\,\,\,\,\,\,and\,\,\,\,\,\,\,\,3x + 6y = - 3\,\,\left( 2 \right) \cr & {\text{Solve the equation }}\left( 1 \right)\,\,{\text{for }}y \cr & y = \frac{{6 - 2x}}{3} \cr & {\text{Substitute }}y{\text{ into the equation 2}} \cr & 3x + 6\left( {\frac{{6 - 2x}}{3}} \right) = - 3 \cr & 3x + 12 - 4x = - 3 \cr & - x = - 15 \cr & x = 15 \cr & y = \frac{{6 - 2x}}{3} \Rightarrow \frac{{6 - 2\left( {15} \right)}}{3} \cr & y = - 8 \cr & {\text{The critical point is }}\left( {15, - 8} \right) \cr & {\text{Use The Second Partials Test at the critical point}} \cr & D = {f_{xx}}\left( {{x_0},{y_0}} \right){f_{yy}}\left( {{x_0},{y_0}} \right) - f_{xy}^2\left( {{x_0},{y_0}} \right) \cr & D\left( {15, - 8} \right) = {f_{xx}}\left( {15, - 8} \right){f_{yy}}\left( {15, - 8} \right) - f_{xy}^2\left( {15, - 8} \right) \cr & D\left( {15, - 8} \right) = \left( 2 \right)\left( 6 \right) - \left( 3 \right) \cr & D\left( {15, - 8} \right) = 9 \cr & D > 0{\text{ and }}{f_{xx}}\left( {{x_0},{y_0}} \right){\text{ > 0}}{\text{, then }}f\left( {x,y} \right){\text{ has a relative mimimun at }}\left( {15, - 8} \right) \cr} $$