How do I do this? Find all values of x and y such that fx(x, y) = 0 and fy(x, y) = 0 simultaneously. f(x, y) = x2 + 4xy + y2 − 34x − 32y + 38

Question

Find all values of x and y such that
fx(x, y) = 0
and
fy(x, y) = 0
simultaneously.
f(x, y) = x2 + 4xy + y2 − 34x − 32y + 38

Answer 1

f_x = 2x + 4y - 34 = 0

f_y = 4x + 2y - 32 = 0

 

2x + 4y = 34

4x + 2y = 32

 

Multiply the first equation by -2 and then add the equations together:

 

-4x - 8y = -68

 4x + 2y = 32

 

So, -6y = -36     y = 6 and x = 5