Question on existence of lower point

Question

Let $X$ be a Banach space, $u,v\in X$ such that $u\ne v$ and $\varphi:X\rightarrow\mathbb{R}$. I would like to know whether there exists $w\in]u,v[$ such that for all neighborhood $W$ of $w$ there exists $\bar w\in W\bigcap]u,v[$ such that $\varphi(\bar w)\leq \varphi(w)$ and $\bar w\ne w$.