Existence of global minimum of convex function

Let $f:I\longrightarrow \mathbb{R}$ a convex function cotinous on a interval $I$.If $f$ not is monotone, then is true that $f$ has a global minimum in inner of $I$?

--My attempt: because $f$ is no monotone exists $x_1<x_2<x_3$ with $f(x_1)>f(x_2)<f(x_3)$.Because $f$ is continuous on $[x_1,x_3]$ exists a point of minimum $x_0$ on $[x_1,x_3]$. The point $x_0$ is distinct from $x_1$ and $x_3$ and $x_0\in\overset{\circ}{I}$.My intuiction says thats $x_0$ is a minimum global of $f$. How can proofs this? It's for exemple true that $f$ is increasing for $x>x_3$ e decreasing for $x<x_1$?