A cubic polynomial in $n$ variables has at most one strict local minimum.

Question

How would you prove the following claim?

A cubic polynomial $p:\Bbb R^n \to \Bbb R$ has at most one strict local minimum.

I have an answer in mind, but I'm wondering if it is trivial or the simplicity and generality of this result surprises you at least a bit.

Note. In contrast, analyzing a degree 4 polynomial already becomes complicated: How many strict local minima a quartic polynomial in two variables might have?

Answer 1

If $x_0, v_0 \in \mathbb{R}^n$, then $t \mapsto p(x_0+v_0t) : \mathbb{R} \to \mathbb{R}$ is a polynomial of degree at most $3$, so it has at most one isolated minimum.

Now assume $p$ has more than one isolated minimum and restrict to the line through any two of them.