Question

LaGrange Multipliers Help Please!

Answer 1

f(x,y,z) = xy + 2xz + 2yz

Constraint xyz = 32

I know the steps of finding the partial derivatives but I get stuck after that. Do I also need to find the Partial derivative of \[\lambda\] - Having 3 variables is proving difficult for me!

Any help as to how I can solve it?

Thanks!

Answer 2

Step 1
Fx & Fy
i.e differentiate with respect to x , Fx ,then differentiate with respect to y, Fy.
Step 2
F x=0 and Fy=0 ( solve simultaneously )

Step 3
Substitute in the constraint to get the critical points (x,y,z).
Solve for x in terms of \[\lambda\].

Answer 3

What about the partial derivative with respect to z?

Answer 4

there are two methods to working this? You can find it also and by finding the z

Answer 5

This is my attempt, does not necessarily have to be correct.
F( x,y,λ) = f(x,y)- λ[ g(x,y)- c]

Fx= y+2z-λyz
Fy= x+2z-λxz

Fy=0
x+2z-λxz=0
λ= x+2z/xz

Fx=0
y+2z-λyz=0
λ= y+2z/yz

solve simultaneously

Answer 6

Continue from here...

Also Fλ= xyz, that why it is not necessary to find it because it give us the constraint, which already given.

Answer 7

How do you solve simultaneously when we habe xz i the first eqn and yz in the second?

Answer 8

but we have λ in both

Answer 9

Yes, and...

Answer 10

lol

Answer 11

λ- (y+2z/yz)=0 equation 1
λ- (x+2z/xz)=0 equation 2

Answer 12

or you could solve by letting λ= λ

Answer 13

Disclaimer, this is how I would attempt this question based on how i was taught

Answer 14

Haha! I'm only kidding!

Answer 15

do u agree or not?

Answer 16

I'll quickly give it a go and see what I get...

Answer 17

well its not quiet... clear so I need to get help