Maximizing C_t with Lagrangian: First Order Condition Explained

[Charlotte87]

Homework Statement

Maximize C_{t} for any given expenditure level

\int_{0}^{1}P_{t}(i)C_{t}(i)di\equiv Z_{t}

The Attempt at a Solution

The Lagrangian is given by:
L = \left(\int_{0}^{1}C_{t}(i)^{1-(1/\varepsilon)}di\right)^{\varepsilon/(\varepsilon-1)} - \lambda \left(\int_{0}^{1}P_{t}(i)C_{t}(i)di - Z_{t}\right)

I know that the first order condition is

C_{t}(i)^{-1/\varepsilon}C_{t}^{1/\varepsilon} = \lambda P_{t}(i) for all i \in (0,1)

But I do not understand how they get to this answer. Can anyone help me?

 

[HallsofIvy]

Do you know the "Euler-Lagrange" equation?
http://en.wikipedia.org/wiki/Euler–Lagrange_equation

 

[Charlotte87]

No, I have to admitt I've never heard of it...