Larange multipliers - please help?

Question

Let f(x,y) = -9 x**2 - 2 x y + y**2 + 5. Use Lagrange multipliers to find the minimum of f subject to the constraint 6 x - y = 9

y = Ax where A=___________

The function f achieves its minimum, subject to the given constraint, where
x =___
y =____
lambda =_____
f =_____

Answer 1

f(x, y) = -9x^2 - 2xy + y^2 + 5
g(x, y) = 6x - y - 9 = 0
L(x, y) = f(x, y) + λg(x, y)
= -9x^2 - 2xy + y^2 + 5 + 6λx - λy - 9λ
∂L/∂x = -18x - 2y + 6λ = 0
∂L/∂y = -2x + 2y - λ = 0
∂L/∂λ = 6x - y - 9 = 0.
So from the first two equations,
3λ = 9x + y = 3(-2x + 2y)
<=> 15x = 5y
<=> y = 3x
Then 6x - y - 9 = 0 => 6x - 3x = 9
=> x = 3, y = 9, λ = -2x + 2y = 12
and f = -9(9) - 2(3)(9) + 9^2 + 5
= -49.

We can tell this is indeed a minimum because there is only one critical point, and the constraint allows us to send both x and y off to either +∞ or -∞, in which case f would go to +∞.