$F(x,y) = x^3 + 3xy - 15x^2-15y^2+72x$ Step 1: $f_x = 3x^2 + 3y - 30x+72$ and $f_y=3x-30y$ $r=f_{xx} = 6x-30$ $t=f_{yy}=-30 \ $and $s=f_{xy} = 3$   Step 2: We now solve, $f_x = 0, \ f_y=0$ as a simultaneous equations. $\therefore 3x^2 + 3y - 30x +72=0$ and $3x-30y=0;$ hence $x=10y$ To eliminate x, we put x=10y in second equation $\therefore 100y^2 - 99y + 24 = 0$ On Solving, $\therefore y=0.565 \text{ or } y=0.424$ when $y=0.565, \ \ x=5.65$ and when $y=0.424, \ \ x=4.24$   Step 3: For $x=5.65, \ \ y=0.565,$ $r= f_{xx}=3.9$ $t=f_{yy} = -30$ $s=f_{xy} = 3$ Now, \(r \ t-s^2 \lt0.\) Hence we reject this pair.   When $x=4.24, \ y=0.424,$ $r=f_{xx} = - 4.56$ $t=f_{yy} = -30$ $s=f_{xy} = 3$ Now, \(r\ t-s^2\gt0.\) Hence, we accept the pair. $x=4.24 $ and $y=0.424$