What is the extreme boundary of A in l-1?

Question

We know that $ A=\{ f \in \ell_1 (N): f(x) \geq 0 , \sum_{x=1}^ \infty f(x)\leq 1, \sum_{x=1}^ \infty \frac{ (-1)^x f(x)}{x}=0\} $ is weak* compact and convex. Why the extreme boundary of $ A$ is $\{0\} \cup \{ f \in A : \sum_{x=1}^\infty f(x) =1\}$ such that $f(x)$ is nonzero for exactly two indices $x$ ? My another question is determining the supremum of $\sum_{x=1}^\infty \frac{f(x)x^2}{2^x}$ over $f \in A$ ?