The graph of the derivative and its sign

Overlaying the graph of the derivative makes the rise and fall of the original function readable at a glance. We set $f(x) = x^3 - 3x$ and $f'(x) = 3x^2 - 3$ side by side.

Sign and monotonicity

Since $f'(x) = 3(x - 1)(x + 1)$, the sign changes as follows.

The points $x = -1$ and $x = 1$, where $f'$ crosses the $x$-axis, are where $f$ has its extrema. The value $f(-1) = 2$ is the local maximum and $f(1) = -2$ the local minimum.

The size of the value is the steepness

The value of $f'$ itself is the slope of the tangent.

The minimum of $f'$ also falls at $x = 0$, which is where the slope of $f$ is smallest: the inflection point.

Symmetry

$f$ is odd, so it has point symmetry about the origin, and $f'$ is even, so it is symmetric about the $y$-axis. Differentiation exchanges even and odd.

Crossings of the $x$-axis

Factoring $f(x) = x(x^2 - 3)$ shows that $f$ meets the $x$-axis at the three points $x = 0$ and $x = \pm\sqrt{3}$. The local maximum is positive and the local minimum negative, so three crossings can also be deduced from the monotonicity alone.

Differentiating once more

We have $f''(x) = 6x$, whose sign changes at $x = 0$. That the concavity of $f$ reverses at the origin and that $f'$ is smallest there are two ways of saying the same thing.

Watch for the change of sign

A point where $f' = 0$ is only a candidate for an extremum. For $f(x) = x^3$ we have $f'(0) = 0$, yet the sign of $f'$ does not change, so it is not an extremum. The change of sign has to be checked as well.

The cubic on the graph is $y = x^3 - 3x$, the parabola is the derivative $y = 3x^2 - 3$, and the large dots are the local maximum $(-1, 2)$ and the local minimum $(1, -2)$.