# Math Help - Lagrange multipliers

1. ## Lagrange multipliers

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).

9. f (x,y,z) = xyz; x^2 + 2y^2 + 3z^2 = 6

I keep going in circles with this question. I get the gradient of f(x) and g(x) and make them equal each other while the gradient of g(x) is multiplied by lambda (the Lagrange multiplier), and then I multiply every equation on the left and right by a variable so that each equation on the left is xyz. I then try to do some algebraic manipulation but since I can't cancel out the sufficient number of variables I can't solve the problem.

To get the gradients you have to take the partial derivatives with respect to x, y and z for both f(x, y, z) = xyz and g(x, y, z) = x^2 + 2y^2 + 3z^2 = 6.

Then you have to multiply the gradient of g(x, y, z) by lambda and make the gradient of f equal the gradient of g (when g is multiplied by lambda).

2. Originally Posted by Undefdisfigure
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).

9. f (x,y,z) = xyz; x^2 + 2y^2 + 3z^2 = 6

I keep going in circles with this question. I get the gradient of f(x) and g(x) and make them equal each other while the gradient of g(x) is multiplied by lambda (the Lagrange multiplier), and then I multiply every equation on the left and right by a variable so that each equation on the left is xyz. I then try to do some algebraic manipulation but since I can't cancel out the sufficient number of variables I can't solve the problem.

To get the gradients you have to take the partial derivatives with respect to x, y and z for both f(x, y, z) = xyz and g(x, y, z) = x^2 + 2y^2 + 3z^2 = 6.

Then you have to multiply the gradient of g(x, y, z) by lambda and make the gradient of f equal the gradient of g (when g is multiplied by lambda).
You form the Lagrangian:

$L(x,y,z,\lambda)=xyz + \lambda (x^2+2y^2+3z^2-6)$

Then your system of equations is:

$
\frac{\partial}{\partial x}L(x,y,x,\lambda)=0
$

$
\frac{\partial}{\partial y}L(x,y,x,\lambda)=0
$

$
\frac{\partial}{\partial z}L(x,y,x,\lambda)=0
$

$
\frac{\partial}{\partial \lambda}L(x,y,x,\lambda)=0
$

(the last of these is just the constraint equation)

Is that what you have tried to do?

RonL

3. Originally Posted by CaptainBlack
You form the Lagrangian:

$L(x,y,z,\lambda)=xyz + \lambda (x^2+2y^2+3z^2-6)$

Then your system of equations is:

$
\frac{\partial}{\partial x}L(x,y,z,\lambda)=0
$

$
\frac{\partial}{\partial y}L(x,y,z,\lambda)=0
$

$
\frac{\partial}{\partial z}L(x,y,z,\lambda)=0
$

$
\frac{\partial}{\partial \lambda}L(x,y,z,\lambda)=0
$

(the last of these is just the constraint equation)

Is that what you have tried to do?

RonL
Yeah, but why did you make the third variable in L, i.e. L(x ,y, x, lambda) x?
Shouldn't it be z?

And I got as far as you've outlined. Now I have to solve for a variable and get a numerical value for a variable so I can solve the remaining ones. How do I do this?

Help plz, thanks.

4. Originally Posted by Undefdisfigure
Yeah, but why did you make the third variable in L, i.e. L(x ,y, x, lambda) x?
Shouldn't it be z?

And I got as far as you've outlined. Now I have to solve for a variable and get a numerical value for a variable so I can solve the remaining ones. How do I do this?

Help plz, thanks.
Sorry should be z - typo and copy and paste there-of.

RonL