Chapter 15 - Section 15.3 - Maxima and Minima - Exercises - Page 1117: 12

$${\text{Relative maximum at the point }}\left( {0,0,4} \right)$$

$$\eqalign{ & f\left( {x,y} \right) = 4 - \left( {{x^2} + {y^2}} \right) \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[ {4 - \left( {{x^2} + {y^2}} \right)} \right] \cr & {f_x}\left( {x,y} \right) = - 2x \cr & and \cr & {f_y}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {4 - \left( {{x^2} + {y^2}} \right)} \right] \cr & {f_y}\left( {x,y} \right) = - 2y \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 = 0,{\text{ }} - 2y = 0 \cr & {\text{Solving the equations, we obtain}} \cr & x = 0,{\text{ }}y = 0 \cr & {\text{The critical point is }}\left( {0,0} \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} \right] = - 2 \cr & {f_{yy}}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ { - 2y} \right] = - 2 \cr & {f_{xy}}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ { - 2x} \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( {0,0} \right) \cr & d\left( {0,0} \right) = {f_{xx}}\left( {0,0} \right){f_{yy}}\left( {0,0} \right) - {\left[ {{f_{xy}}\left( {0,0} \right)} \right]^2} \cr & d\left( {0,0} \right) = \left( { - 2} \right)\left( { - 2} \right) - {\left[ 0 \right]^2} \cr & d\left( {0,0} \right) = 4 \cr & d > 0,{\text{ and }}{f_{xx}}\left( {0,0} \right) = - 2 < 0 \cr & {\text{then}} \cr & f\left( {x,y} \right){\text{ has a relative maximum at }}\left( {0,0,f\left( {0,0} \right)} \right) \cr & f\left( {0,0} \right) = 4 - \left( {{0^2} + {0^2}} \right) \cr & f\left( {0,0} \right) = 4 \cr & {\text{Relative maximum at the point }}\left( {0,0,4} \right) \cr} $$