# Lagrange multipliers

1. Apr 20, 2010

### nhartung

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. 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:

fx = 16x - 8x$$\lambda$$ = 0
fy = 4z - 2y$$\lambda$$ = 0
fz = 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?

2. Apr 20, 2010

### nhartung

ah nevermind it checks if I use x = 0.