Polynomial function/Function behavior from differentiability/Fact

A real polynomial function

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

of degree ${}d\geq 1$ has at most ${}d-1$ local extrema, and one can partition the real numbers into at most ${}d$ intervals, on which ${}f$ is alternatingly strictly increasing or strictly decreasing.

Write a proof