# Lagrange multipliers and partial derivatives

1. Dec 13, 2009

### phrygian

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

Find the point on 2x + 3y + z - 11 = 0 for which 4x^2 +y^2 +z^2 is a minimum

2. Relevant equations

3. The attempt at a solution

Using lagrange multipliers I find:

F = 4x^2 + y^2 + z^2 + l(2x + 3y + z)

Finding the partial derivatives I get the three equations:

df/dx = 8x + 2l
df/dy= 2y + 3l
df/dz= 2z + l

This is where I am stuck, what are the next steps for solving the system of equations?

2. Dec 13, 2009

### HallsofIvy

Well, first set them equal to 0! They are equivalent to $x= -\lambda$, $2y/3-\lambda$ and $2z= -\lambda$. You can easily eliminate $\lambda$ by setting those equal to each other. And don't forget that you have 2x+ 3y+ z= 11 as a third equation.