Find A and B from a local minimum value

Question

Find A and B so that f(x,y) = x² + Ax + y² + B had a minimum value of 19 at (1,0).

Answer 1

f(x,y) = x² + Ax + y² + B

∂f/∂x = 2x + A

∂f/∂y = 2y

At (1,0):

∂f/∂x = 2(1) + A = 2 + A

∂f/∂y = 2y = 2(0) = 0

Set partial derivatives equal to each other since they need to both be 0 anyway:

2 + A = 0

A = -2

Hence, we have:

f(1,0) = 1² + (-2) + 0² + B = 1 - 2 + B = B - 1

Since the multivariable function is said to have a minimum of 19 at (1,0), then we have:

B - 1 = 19

B = 20

Thus, A and B are -2 and 20 respectively, making the final function f(x,y) = x² - 2x + y² + 20

Hope this helped!