Chapter 13 - Partial Derivatives - Chapter 13 Review Exercises - Page 998: 36

$${\text{relative minimum for all points }}\left( {x,\frac{{2x}}{3}} \right)$$

$$\eqalign{ & f\left( {x,y} \right) = 4{x^2} - 12xy + 9{y^2} \cr & \,\,\,\,{f_x}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {4{x^2} - 12xy + 9{y^2}} \right] \cr & \,\,\,\,\,{f_x}\left( {x,y} \right) = 8x - 12y \cr & \,\,\,\,\,{f_{xx}}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {8x - 12y} \right] \cr & \,\,\,\,\,{f_{xx}}\left( {x,y} \right) = 8 \cr & and \cr & \,\,\,\,\,{f_y}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {4{x^2} - 12xy + 9{y^2}} \right] \cr & \,\,\,\,\,{f_y}\left( {x,y} \right) = - 12x + 18y \cr & \,\,\,\,\,{f_{yy}}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ { - 12x + 18y} \right] \cr & \,\,\,\,\,{f_{yy}}\left( {x,y} \right) = 18 \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[ {8x - 12y} \right] \cr & \,\,\,\,\,{f_{xy}}\left( {x,y} \right) = - 12 \cr & \cr & {\text{Setting the first partial derivatives to 0}} \cr & {f_x}\left( {x,y} \right) = 8x - 12y = 0{\text{ and }}{f_y}\left( {x,y} \right) = - 12x + 18y = 0 \cr & \,\,\,\,8x - 12y = 0 \cr & \,\,\,\,\,\,\,x = \frac{3}{2}y \cr & {\text{Substitute }}\frac{3}{2}y{\text{ for }}x{\text{ into }} - 12x + 18y = 0 \cr & \,\,\,\,\,\, - 12\left( {\frac{3}{2}y} \right) + 18y = 0 \cr & \,\,\,\,\, - 18 + 18y = 0 \cr & \,\,\,\,\,\,\,0 = 0 \cr & {\text{Infinite solutions}}{\text{, all the points }}t{\text{ for }}x = \frac{3}{2}y{\text{ are solutions}}{\text{.}} \cr & x = \frac{3}{2}y \Rightarrow y = \frac{{2x}}{3},\,\,\,{\text{for }}x = t,\,\,\,\,y = \frac{{2t}}{3} \cr & {\text{Then}}{\text{, The critical points are }}\left( {t,\frac{{2t}}{3}} \right) \cr & {\text{Use The Second Partials Test at the critical points}} \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( 8 \right)\left( {18} \right) - {\left( { - 12} \right)^2} \cr & D = 2 \cr & {\text{and}} \cr & {f_{xx}}\left( {x,y} \right) = 8 > 0 \cr & \cr & D > 0{\text{ and }}{f_{xx}} > 0,{\text{ There is a relative minimum at }}\left( {t,\frac{{2t}}{3}} \right) \cr & or \cr & {\text{There is a relative minimum for all points }}\left( {x,\frac{{2x}}{3}} \right) \cr} $$