# Thread: Lagrange Multipliers

1. ## Lagrange Multipliers

Hi, am studying for final exam and has been long time since did lagrange and am stumped on the following,

Using method of LM find the maximum and minimum values of the function;

f(x,y,z) = x

subject to the constraints (x^2) + 2(y^2) + 2(z^2) = 8
and x + y = z

I used LM to get the following;

(1,0,0) = a(2x, 4y, 4z) + b(1, 1, -1)

Where a = lambda1 and b = lambda 2 (lagrange multipliers)

but i can't solve the equations to give max/min?

Any help would be greatly appreciated.
Cheers.

2. You have the 3 equations :

i 1 = 2ax + b

ii 0 = 4ay + b

iii 0 = 4az - b

adding ii and iii we get y = - z using this in the constraint x+y=z we
obtain:
x -z = z or x=2z

Using this in the other constraint x^2 +2y^2 +2z^2 = 8

We obtain:

4*z^2 + 2z^2 +2z^2 = 8

z= +1 , x= + 2 , y = -1,1 (2,-1,1) and (-2,1,-1) are candidates for max and min

Max is 2 Min is -2 since f(x,y,z) = x

3. ## Alternative approach

Alternative approach:

i 1 = 2ax + b

ii 0 = 4ay + b

iii 0 = 4az - b

adding ii and iii we get y = - z using this in the constraint x+y=z we
obtain:
x -z = z or x=2z

using i and iii we get b= 1/2

Then:

x= 1/4a y = -1/8a z= 1/8a

Using the constraint x^2 +y^2 +z^2 = 8

a = + 1/8

a= 1/8 yields (2,-1,1)

a= -1/8 yields (-2,1,-1)