Let $X$ be a Banach space. And let $X^* $ be the dual space of $X$. Let $E_X$ and $E_{X^*}$ denote the extreme points of the unit ball of $X$ and $X^*$. Let $x\in X$ and $|f(x)|=1$ for every $f\in E_{X^*}.$ Does that imply $x\in E_X?$

Can anyone suggest a text to study theory of extreme points of a convex set in a Banach space(for beginners)?