If $f$ is quasiconvex and differentiable then $f$ is convex?

Question

Let $f\colon C \to \mathbb{R}$ be a homogeneous function of degree $1$, $C\subset \mathbb{R}^n$ a cone. If $f$ is differentiable then it is true: if $f$ is quasiconvex $\Rightarrow$ $f$ is convex.