Find A and B that f(x, y) has a local min at point

Question

Find A and B so that f(x, y)=x^2 + Ay + y^2 + B has a local minimum at the point (0,8), with z-coordinate 10.

Answer 1

complete the square: f(x,y) = x^2 + Ay + y^2 + B = (x - 0)^2 + (y+A/2)^2 - A^2/4 + B

The min is at (0, -A/2), so if it is at (0, 8), -A/2 = 8, or A = -16

f(0,8) = (x - 0)^2 + (y+A/2)^2 - A^2/4 + B = 0 + 0 - 16^2/4 + B = -64 + B

As f(0,8) = 10, -64 + B = 10, or B = 74

A = -16; B = 74

f(x,y) = x^2 - 16y + y^2 + 74