Lagrange Multipliers: Find Max of 8x2 + 4yz - 16z + 600

[nhartung]

Homework Statement

Assume that the surface temperature distribution of an ellipsoid shaped object given by 4x² + y² + 4z² = 16 is T(x,y,z) = 8x² + 4yz - 16z + 600.

Homework Equations

The Attempt at a Solution

I'm assuming we just have to find the maximum value of this function using the lagrange method.

I started by writing the equation like this:

8x² + 4yz -16z + 600 - 4x²$$\lambda$$ - y²$$\lambda$$ - 4z²$$\lambda$$ + 16$$\lambda$$.

Then I found the 4 partials and set them to 0:

f_x = 16x - 8x$$\lambda$$ = 0
f_y = 4z - 2y$$\lambda$$ = 0
f_z = 4y - 16 - 8z$$\lambda$$ = 0
f_$$\lambda$$ = -4x² - y² - 4z² + 16 = 0

My problem comes next when I try to solve this system of equations.
When I solve them I get:
x = 1 (or 0?)
y = z = -(4/3)
$$\lambda$$ = 2

These don't check out.

Does it looks like I'm going about this problem correctly? If so what am I doing wrong when solving the system of equations?

 

[nhartung]

ah nevermind it checks if I use x = 0.