How Do You Solve a Lagrange Multiplier Problem with a Spherical Constraint?

[Elliotc]

1. Assume we have function V(x,y,z) = 2x2y2z = 8xyz and we wish to maximise this function subject to the constraint x^2+Y^2+z^2=9. Find the value of V at which the max occurs



2. Function: V(x,y,z) = 2x2y2z = 8xyz
Constraint: x^2+Y^2+z^2=9



3. So far I have gone
Φ= 8xyz + λ(x^2+y^2+z^2 - 9)

∂Φ/∂x = 8yz + 2λx = 0 equation 1
∂Φ/∂y = 8xz + 2λy = 0 equation 2
∂Φ/∂z = 8xy + 2λz = 0 equation 3
x^2 + y^2 + z^2 = 9

so then I have multiplied equation 1 by x, 2 by y, and 3 by z

so I am left with
8xyz + 2λx^2 = 0
8xyz + 2λy^2 = 0
8xyz + 2λz^2 = 0

add these together and
24xyz +2λ(x^2 + y^2 + z^2)
I know that x^2 + y^2 + z^2 =9 and v=8xyz so
3V=-18λ
V=-6λ

Now I have no idea how to go about the next stage, I am struggling to rearrange the equations to get an answer.

 

[Ray Vickson]

1. Assume we have function V(x,y,z) = 2x2y2z = 8xyz and we wish to maximise this function subject to the constraint x^2+Y^2+z^2=9. Find the value of V at which the max occurs


2. Function: V(x,y,z) = 2x2y2z = 8xyz
Constraint: x^2+Y^2+z^2=9



3. So far I have gone
Φ= 8xyz + λ(x^2+y^2+z^2 - 9)

∂Φ/∂x = 8yz + 2λx = 0 equation 1
∂Φ/∂y = 8xz + 2λy = 0 equation 2
∂Φ/∂z = 8xy + 2λz = 0 equation 3
x^2 + y^2 + z^2 = 9

so then I have multiplied equation 1 by x, 2 by y, and 3 by z

so I am left with
8xyz + 2λx^2 = 0
8xyz + 2λy^2 = 0
8xyz + 2λz^2 = 0

add these together and
24xyz +2λ(x^2 + y^2 + z^2)
I know that x^2 + y^2 + z^2 =9 and v=8xyz so
3V=-18λ
V=-6λ

Now I have no idea how to go about the next stage, I am struggling to rearrange the equations to get an answer.

Your equations 8xyz + 2*lambda*x^2 = 0, etc., imply that x^2, y^2 and z^2 are equal, so it is easy to get them. Of course, x, y and z are thus determined only up to a +- sign, so you still need to think a bit about which choices make sense.

RGV