Find the absolute minimum and maximum of F(x,y,z) = x2 - 2x - y2 + z2 on the ellipsoid G(x,y,z) = x2 + 4y2 + z2 = 4
The Attempt at a Solution
I was thinking of trying to solve this by using Lagrange multipliers. So, finding the gradients:
Fx = 2x - 2 = Gx = λ 2x
Fy = - 2y = Gy = λ 8y
Fz = 2z = Gz = λ 2z
From the first partial derivative I have 2x - 2 - λ2x = 0, which suggests x = 1/(1-λ). From the second partial derivative I have y(-2 - λ * 8) = 0, which suggests y = 0. Similarly, from the third partial derivative I have z(2 - λ * 2) = 0, which suggests z = 0. From G(1/(1-λ),0,0) I get λ = ((-/+) 1/2) + 1, or λ = 1/2 or 3/2.
Therefore, x = -2 or 2. Evaluating F(-2,0,0) = 8 and F(2,0,0) = 0. So, (-2,0,0) is our max and (2,0,0) is our min.
Does that sound about right?