Find the maximum value of 2x+3y+7z, subject to the constraint x^4+y^4+z^4= 1.
First I formed my Lagrangian for this problem:
L(x,y,z) = 2x + 3y + 7z - λ(x^4 + y^4 + z^4 -1)
I then took my partial derivatives:
dL/dx = 2 - 4λx^3
dL/dy = 3 - 4λy^3
dL/dz = 7 - 4λz^3
For each of these I made x,y and z the subject respectively, giving:
x^3 = 1/2λ ; y^3 = 3/4λ ; z^3 = 7/4λ
I now subbed these into my constraint. This gave a scary equation with the above λ expressions raised to the power of 4/3 being equal to 1. This is where I'm stuck, can anyone help me please?
I found a solution, but not in surd form, 2.38534589 satisfies the λ equation being equal to 1. But my answer must be in the form of a surd
It would have made more sense if you had made the subject of each equation, the set them all equal: . Or, you can eliminate by dividing one equation by another. From and , we get so that and from and we get so . Now so and so . You can put those into the constraint equation to get a single equation for y, and then get x and z.
Really need to start checking what you've posted before I post Matthew looks like what I've done thus far is right. I just thought it was a bit too complicated. It's far more complicated than the one on the examples sheet we were given, because the constraint there was only :\
Hello, mitch_nufc!
I would solve forFind the maximum value of , subject to the constraint
First I formed my Lagrangian for this problem:
I then took my partial derivatives:
. .
We have: . .[1]
We have: . .[2]
Substitute [1] and [2] into the constraint: .
Factor: .
. .
And we can back-substitute to determine and