A question about Lagrange optimization.

A question about Lagrange optimization.

This is a statement from a finance textbook - I find it pretty clear
everywhere else, but this particular part I am clueless. Hopefully you
guys can figure it out.
The problem is solving: $$\sup_{(C_t,M_t)} E[\int_0^\infty e^{-\delta t}
\, u(C_t,M_t/I_t) \,dt]$$ $$s.t. E[ \int_0^\infty \zeta_t
(C_t+\frac{M_t}{I_t}r_t)]\leq W_0$$
where $W_0, \delta$ are constants. u is some function (as nice as it has
to be) and in general $(C_t) , (I_t), (M_t) , (r_t)$ are all stochastic
processes. It is just stated, without further ado, that the first order
conditions are $$ e^{-\delta t} u_C(C_t,M_t/I_t) = \psi \zeta$$ $$
e^{-\delta t} u_M(C_t,M_t/I_t) = \psi \zeta r_t$$ where $\psi$ is a
Lagrange multiplier, which is set such that the constraint holds. $U_C$
and $U_M$ are derivatives with respect to C and M respectively. Then he
continuous using this result as a truth and never speaks of it again.
To be honest I've got no idea - I've seen lagrange used before but I don't
see it giving me anything here. (p.s. sorry - I couldn't think of a better
title, feel very free to change it)