Polynomial function/Function behavior from differentiability/Fact/Proof/Exercise

Prove that a real polynomial function

$f\colon \mathbb {R} \longrightarrow \mathbb {R}$

of degree ${}d\geq 1$ has at most ${}d-1$ extrema, and moreover the real numbers can be divided into at most ${}d$ sections, where ${}f$ is strictly increasing or strictly decreasing.

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