- #1
Elliotc
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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.
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.