Let $X$ be a normed space and $x_0,y_0\in X$. How to find a member $x^*$ of $X^*$ whose kernel is the line through $x_0$ and $-x_0$ such that $x^*y_0\geq 0$?
I think we should use Hahn-Banach extension theorem and its corollaries: Since the line through $x_0$ and $x_0$, called $L$, is a closed subspace of the normed space $X$ so if $y_0\notin L $ then there is a $x^*\in X^*$ such that $L\subset \ker x^*$ and $x^*y_0=\mathrm{dist}(y_0,L)$. But this doesn't imply $L=\ker x^*$!
Unless $X$ is a $2$-dimensional space no such functional exists.
$L$ is a $1$-dimensional subspace of $X$ but whenever $f$ is a bounded linear functional on $X$ we have that $\operatorname{codim}(\ker f) = 1$. It is then easy to see that $\dim L = 1$ and $\dim X/L = 1$ together imply that $\dim X = 2$.