% [** F1: \small?]
\begin{itemize} % [** F1: Presumably started on previous page]
\item[]
of coordinates, we see that the equation $f(x)=0$ becomes a reduced cubic equation
$X^3 + pX + q = 0$ (§~42).

\item[6.] Find the inflexion  tangent to $y = x^3 + 6x^2 - 3x + 1$ and transform
$x^3 + 6x^2 - 3x + 1 = 0$ into a reduced cubic equation.
\end{itemize}
% [** F1: End of \small]

\paragraph{60. Real Roots of a Real Cubic Equation.} It suffices to consider
\[
f(x) = x^3 - 3lx + q \qquad (l \neq 0), % [** F1: Not pushing to right margin]
\]
in view of Ex.~5 above. Then $f' = 3 (x^2 - l)$, $f'' = 6x$. If $l<0$, there is
no bend point and the cubic equation $f(x)=0$ has a single real root.

If $l>0$, there are two bend points
\[
(\sqrt{l}, q - 2l\sqrt{l}),\qquad    (-\sqrt{l}, q + 2l\sqrt{l}),
\]
which are shown by crosses in Figs.~18--20 for the graph of $y=f(x)$ in the
%
% [** Illustrations]
% \caption{$q \geq  2l\sqrt{l}$} FIG. 18
% \caption{$q \leq -2l\sqrt{l}$} FIG. 19
%
three possible cases specified by the inequalities shown below the figures.
For a large positive~$x$, the term $x^3$ in $f(x)$ predominates, so that the graph
contains a point high up in the first quadrant,
thence extends downward to the
right-hand bend point, then ascends to
the left-hand bend point, and finally descends.
As a check, the graph contains
a point far down in the third quadrant,
since for $x$ negative, but sufficiently large
numerically, the term $x^3$ predominates and the sign of~$y$ is negative.
%
% [** Illustration]
% \caption{$-2l\sqrt{l} < q < 2l\sqrt{l}$}  Fig. 20
%

If the equality sign holds in Fig.~18 or Fig.~19, a necessary and sufficient
condition for which is $q^2 = 4l^3$, one of the bend points is on the $x$-axis, and
the cubic equation has a double root. The inequalities in Fig.~20 hold
if and only if $q^2 < 4l^3$, which implies that $l>0$. Hence $x^3 - 3lx + q = 0$
\emph{has three distinct real roots if and only if $q^2 < 4l^3$, a single real root if and
only if $q^2 > 4l^3$, a double root (necessarily real) if and only if $q^2 = 4l^3$ and $l\neq 0$,
and a triple root if $q^2 = 4l^3 = 0$}.
