Partial Differentiation

Question

The Function f(x,y)= x^2+y^2+xy+9x has one critical point. Find it and identify it as a local max local min or saddle point

Answer 1

Finding the partial derivatives we obtain the following f_x=2x+y+9, f_y=2y+x and so the critical point is the solution of the system 2x+y+9=0 and 2y+x=0 which is the point (-6,3). By the derivative test we have that it is a local minimum since D=(f_xx)(f_yy)-(f_xy)^2=3>0 and f_xx>0.