1. What is the difference between f'(x)=0 and f'(xsubzero)=0

2. Solving constrained optimization problem using Lagrangian method:
a.) U = xsub1xsub2 + 2xsub1, s.t. 4xsub1 + 2xsub2 = 60
b.) U = xy, s.t. x + y = 6

3. Find Partial Derivatives for:
a.) y = f(xsub1, xsub2) = 2ln(3xsub1) - 4ln(2xsub1xsub2)


Any help is much appreciated. Thanks for reading!

This post was edited by CamelFinger on Sep 12 2015 07:56pm

f'(xsubzero)= 0 just means the derivative at the initial value of x in the domain is equal to 0 while f'(x)=0 means the derivative is equal to 0 for all x in the domain of f

2
Let L ( x₁ ; x₂ ; λ ) = U ( x₁ ; x₂ ) + λ.( 4x₁ + 2x₂ - 60 )

L = x₁ . x₂ + 2x₁ + λ.( 4x₁ + 2x₂ - 60 )

Express the partial derivatives of L :

∂L / ∂x₁ = x₂ + 2 + 4λ
∂L / ∂x₂ = x₁ + 2λ
∂L / ∂λ = 4x₁ + 2x₂ - 60

Now solve for x₁ , x₂ , λ :
∂L / ∂x₁ = 0
∂L / ∂x₂ = 0
∂L / ∂λ = 0

x₁ = 8
x₂ = 14
λ = - 4

3

Notice that, due to logarithm algebraic properties :
y ( x₁ ; x₂ ) = 2.ln3 - 4.ln2 - 2.lnx₁ - 4.lnx₂

Hence,
∂y / ∂x₁ = - 2 / x₁
∂y / ∂x₂ = - 4 / x₂