Clarifying derivation of Water-filling problem from Convex Optimization

Question

From Boyd & Vandenberghe's Convex Optimization:

How did we obtained the highlighted part? I assume it is derived from inequalities above it, but $\lambda_i$ is not in the highlighted part.

Also, here we have treated $x \succeq 0$ as one of the constraints. Isn't that implicit and not included?, I guess x is not bounded to b e+ive!

Answer 1

Solve $$-1/(\alpha_i+x_i^*)-\lambda_i^*+\nu^*=0$$ for $\lambda_i^*$, yielding $$\lambda_i^*=\nu^*-1/(\alpha_i+x_i^*).$$ Now substitute for $\lambda_i^*$ in $\lambda_i^* x_i^*$.