Definition: Let $X$ be a Banach space. 
We say that a bounded linear functional $x^*:X\to \mathbb{R}$ is an **extreme point of** $B_X = \{x\in X:\|x\|_X\le 1\}$ if whenever 
$$x^* = \frac{1}{2}(y_1^*+y_2^*)$$
for some $y_1^*,y_2^*$ with $\|y_i^*\|_{X^*} \leq 1$ for all $i=1,2,$ we have $x^* = y_1^*=y_2^*.$

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> **Question**: Given a Banach space $X.$
Is it true that there exists an bounded linear extreme functional $x^*:X\to\mathbb{R}$ such that 
$$x^*(x) = 1$$ 
for some $x\in X$ with $\|x\|\leq 1?$

In other words, is it true that every Banach space has at least one extreme functional that is normed by some point?