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?