Let $\mathbf{v}_1,\dots,\mathbf{v}_r$ be vectors in a Euclidean space $\mathbf{V}$.
Let $f \colon \mathbf{V} \to \mathbb{R}$ be a linear function. Prove that $f$ has both a maximum and a minimum value on $\mathrm{Conv}(\mathbf{v}_1,\dots,\mathbf{v}_r)$, and show that it has both a maximizer and a minimizer in $\{\mathbf{v}_1,\dots,\mathbf{v}_r\}$.
Not sure how to prove this so any help would be appreciated.