I could use some explanation for the following problem. How exactly does he solve the system of equations and get 2 different cases from it? Also, how is the discriminant used? Thank you

3. (10 points) Let f(x, y) = x y + 23 - 2zy +6. (a) Find the critical points of f and classify them as local maxima, local minima, saddle points, or none of these. 58 x = 2 xy - Zy = 0 Case i ) y= 0 , x - 2X= 0 50 * = S 2X - 2= 0 SX= 1 ( x2 + 4y- 2X= 0 ( y = /4 critical points : (0, 0 ), ( 2 , 0 ), ( 1, /4 ) discriminant D= 8xx byy - Pay? = 24.4 - (2x-2 ) at ( 0 , 0 ) , DO & 8 x x 70 - local minimum (b) Does f have a global maximum or minimum? Justify your answer. lim $ ( x , 1 ) = 00 Rim 8 ( x , - 1 ) = - 00 X - - DO So there is no global maximum on minimum 2