Using the graph of
You are looking at a graph of f '(x), which shows all the slopes of f(x).
A local minimum on f(x) is located where the slopes of f(x) change from negative to positive.
A local maximum on f(x) is located where the slopes of f(x) change from positive to negative.
An inflection point on f(x) is where concavity changes, and concavity is a determined by the slope of the the slopes.
x = −3: local minimum
x = 1: local maximum
x = −1: inflection point