Here's the Exercise 3.30 from the textbook of Görtz and Wedhorn:
Let $k$ be a field, and let $A$ be a local $k$-algebra of finite type. Prove that $\operatorname{Spec} A$ consists of a single point, and that $A$ is finite-dimensional as a $k$-vector space. In particular $A$ is a local Artin ring (why?), and $\kappa (A)/k$ is a finite field extension.
I see how this follows from some well-known results:
Every finitely generated algebra over a Jacobson ring is Jacobson, so that $A$ is Jacobson. As $A$ is local, this amounts to saying that its maximal ideal is the only prime ideal.
In general, for every maximal ideal $\mathfrak{m}$ in a finitely generated $k$-algebra $A$, the extension $\kappa (\mathfrak{m})/k$ is finite (which follows again from the general results about Jacobson rings).
A ring is Artinian iff it is Noetherian and every prime ideal is maximal, which is the case here.
A finitely generated $k$-algebra is Artinian iff it is finite (Atiyah-Macdonald, Exercise 8.3).
However, all this seems to be an overkill, and I think the authors had in mind some direct argument for the very special case when $A$ is local. Do you see one?