I conjecture following statement in the Banach space $X$ with norm $\|.\|$.

Let $B \subseteq X$ be the unit closed ball in $X$ then,

$\|.\|$ is smooth, in sense $\|.\|$ is differentiable at any nonzero point in $X$, if and only if for any point, say $ x_0 \in \text{bd}(B)$ there is a unique $f \in X^*$ such that $\|f\|=f(x_0).$

My thought: I think proving Left $\rightarrow$ Right is easy It is just a separation, and using smoothness of norm we can show that $f\in X^*$ is unique, but I am interested see your rigorous proof. I don't have any idea about the other way! Even I am not sure it is true! However I am quite positive it is true when $X$ is finite dimension, still any proof in this particular case would be appreciated . (I first assumed $X$ is a reflexive,(I don't know why I did that! maybe because my unexplainable intuition) anyway you may assume reflexivity)

Thanks for nice comments and answers inadvance!