Horizontal and non Horizontal Points of Inlexion

A stationary point
$(x_0, f(x_0))$
is a point of inflexion on a curve
$f(x)$
if
$\frac{d^2f}{dx^2} \|_{x=x_0}=0$
.
If also
$\frac{df}{dx} \|_{x=x_0}=0$
the point is a horizontal point of inflexion.
Example:
$f(x)=x^3(x-1)=x^4-x^3$
.
$\frac{df}{dx}=4x^3-3x^2$
so
$\frac{df}{dx}=0$
at
$x=0$
$\frac{d^f}{dx^2}=12x^2-6x$
so
$\frac{d^2f}{dx^2}=0$
at
$x=0$
.
The point
$(0,f(0))=(0,0)$
is a stationary point of inflexion.

Eample:
$f(x)=x(x^2+2)=x^3+2x$
.
$\frac{df}{dx}=3x^2+2, \: \frac{d^2f}{dx^2}=6x$
.
When
$x=0$
,
$\frac{df}{dx} \neq 0, \: \frac{d^2f}{dx^2} = 0$
.
When
$x=0$
, the point is a point of inflexion but not a stationary point of inflexion.