Maximal distance of a segment from $0$

Pick two distinct vectors $a,b$, with $b\neq 0$, in a Banach space $X$ (not necessarily Hilbert), and define the function $f: [0,1] \to \mathbf{R}$ by $$ \forall t \in [0,1], \quad f(t):=\|a+tb\|. $$ Is it true that $f$ is maximized in $t=0$ or in $t=1$?

I know that $f$ is a continuous function defined on a compact set, hence a maximum exists, let us say in $t_0$. Where is the contradiction if $t_0 \in (0,1)$? I think that I am missing something trivial.

Solution 1:

Closed balls in a normed space are convex. Since $\overline B(0,\max\{\lVert b+a\rVert,\lVert a\rVert\})$ contains both the endpoints of the segment, it must contain the entire segment. Therefore the answer is yes: one of the extremal points maximizes the norm. It is possible (see non-strictly convex norms) that all the points in the segment do, though.