Question

Using the method of Lagrange mulitpliers, optimize the function, f(x,y,z)=x^2+y^2+z^2, subject to x+3y-2z=12.

Answer 1

\[\large f(x,y,z)=x^2+y^2+z^2\]\[\large g(x,y,z)=x+3y-2z=12\]Mmm so it's been a few weeks since I've done these problems.. trying to remember.. We set them up likeeee, this\[\huge \nabla f=\lambda \cdot \nabla g\]So if we take the partial of both sides with respect to X, we get this equation,\[\large f_x=\lambda \cdot g_x\]\[\large 2x=\lambda\]If we continue with the partials, taking with respect to y, then z, we'll get a system of equations.\[\large 2x=\lambda\]\[\large 2y=3\lambda\]\[\large 2z=-2\lambda\]\[\large 12=x+3y-2z\]

Answer 2

So we have a system of 4 equations, and 4 unknowns (x, y, z, lambda).
There are a few different ways to solve systems.

But for this one, I think we can easily solve for x y and z in terms of lambda, and then plug that into the constraint to get a value for lambda.\[\large 2x=\lambda \qquad \rightarrow \qquad x=\frac{1}{2}\lambda\]\[\large 2y=3\lambda \qquad \rightarrow \qquad y=\frac{3}{2}\lambda\]\[\large 2z=-2\lambda \qquad \rightarrow \qquad z=-\lambda\]Plugging these into the constraint gives us,\[\large 12=x+3y-2z \qquad \rightarrow \qquad 12=\left(\frac{1}{2}\lambda\right)+3\left(\frac{3}{2}\lambda\right)-2(-\lambda)\]

Answer 3

THANK YOU!!

Answer 4

lol :D can you figure the rest out from there?

Answer 5

ya all i have to do is solve for (lambda) and then plug it into this-->∇f=λ⋅∇g and solve for x,y, and z? wait how do i solve for x,y and z?

Answer 6

Ah sorry I'm back :O
So did you find your lambda value yet?

Answer 7

ya it's 12/7

Answer 8

\[\large x=\frac{1}{2}\lambda \qquad \rightarrow \qquad x=\frac{1}{2}\left(\frac{12}{7}\right)\]This will get you your x value, and so on :D

Answer 9

After you get an x,y,z.. plug them into f to get a function value.

The problem with these constraint problems, is that it can be hard to tell if you found a maximum, minimum or whatever else :P

But it looks like it doesn't matter, they simply said "Optimize", they didn't want you to classify it any further it looks like.

Answer 10

Hopefully we did that correctly :3 do you have a book answer you can compare it to?

Answer 11

No :( i don't have a book answer but what does it mean by optimize?
i don't understand that.. i mean okay so i get the f(x,y,z) value and is that the answer?

Answer 12

Umm maybe you remember back to calc 1, dealing with optimization.

Like... finding the greatest possible area of a box, given a constraint on the perimeter.

Orrrrr like.. find the dimensions of a box that minimize the cost to create the box, given a constraint on the volume.

In the problem we worked on, we optimized our function when given a certain constraint. We don't actually know if this is a local minimum or maximum of f. There's probably something that could be done to determine that, but we don't need to worry about it. :D

Answer 13

If you solve for x,y,z I think you get these values, and so we would write our answer as,\[\large f\left(\frac{6}{7},\;\frac{18}{7},\;-\frac{12}{7}\right)=\]And ... then whatever value you get :3

Answer 14

okay that's what i did haha
do you mind helping me understand two more problems from calc 3?

Answer 15

I can try :3

Answer 16

kk \[\int\limits_{0}^{4}\int\limits_{(x/2)}^{2}e^ydydx\]

sketch the region and interchange the limits of integration

Answer 17

So our boundaries are,
x=0
x=4

and

y=x/2
y=2

Let's see if we can sketch that effectively.