Constrained optimization problem.

Question

Given constants a > 0, b > 0, c > 0, find the minimum of J = x + y + z subject to (a/x) + (b/y) + (c/z) = 1.

Answer 1

I am assuming that your equation a/x + b/x + c/x = 1 is a typo and you mean to have a/x + b/y + c/z = 1

l(x,y,z) = x + y + z + λ(a/x + b/y + c/z)

Then, ∂l/∂x = 1 - λa/x²

∂l/∂y = 1 - λb/y²

∂l/∂z = 1 - λc/z²

^{Then, if} ∂l/∂x = ∂l/∂y = ∂l/∂z = 0,

Then, x = √(λa), y = √(λb), and z = √(λc)

As a/x + b/y + c/z = 1,

a/√(λa) + b/√(λb) + c/√(λc) = 1

1/√λ (√a + √b + √c) = 1

Then, √λ = (√a + √b + √c)

Thus,

x = √a (√a + √b + √c)

y = √b (√a + √b + √c)

z = √c (√a + √b + √c)