Maximize (nonlinear programming)

[hsong9]

Homework Statement

Maximize xy²z³
subject to x³ + y² + z = 39 and x,y,z > 0

Homework Equations

The Attempt at a Solution

I know the answer ( I got this from some site)

x³ + y² + z = x³ + y²/3 + y²/3 + y²/3 + z/9 + z/9 + z/9 + z/9 + z/9 + z/9 + z/9 + z/9 + z/9 ...
= c(xy²z³)^3/13

But I do not know how I can determine a constant multiple of a suitable power of (xy²z³).

Anybody help me

 

[mighty2000]

?Hello, as a scientist, I would suggest approaching this problem using the method of Lagrange multipliers. This method allows us to find the maximum or minimum value of a function subject to certain constraints. In this case, our function is xy2z3 and our constraint is x3 + y2 + z = 39.

First, we can rewrite our constraint as a function: g(x,y,z) = x3 + y2 + z - 39 = 0.

Next, we can define our Lagrangian as L(x,y,z,λ) = xy2z3 + λ(x3 + y2 + z - 39). λ is our Lagrange multiplier.

We can then take the partial derivatives of L with respect to x, y, z, and λ and set them equal to 0:

∂L/∂x = y2z3 + 3λx2 = 0
∂L/∂y = 2xyz3 + 2λy = 0
∂L/∂z = 3xy2z2 + λ = 0
∂L/∂λ = x3 + y2 + z - 39 = 0

Solving these equations simultaneously will give us the values of x, y, z, and λ that maximize xy2z3 subject to the constraint x3 + y2 + z = 39.

I hope this helps! Let me know if you have any further questions.