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## Homework Statement

Find the absolute minimum and maximum of F(x,y,z) = x

^{2}- 2x - y

^{2}+ z

^{2}on the ellipsoid G(x,y,z) = x

^{2}+ 4y

^{2}+ z

^{2}= 4

## Homework Equations

## The Attempt at a Solution

I was thinking of trying to solve this by using Lagrange multipliers. So, finding the gradients:

F

_{x}= 2x - 2 = G

_{x}= λ 2x

F

_{y}= - 2y = G

_{y}= λ 8y

F

_{z}= 2z = G

_{z}= 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?