of coordinates, we see that the equation f(x)=0 becomes a reduced cubic equation
X^3 +pX+q = 0 (§ 42).

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.

60. Real Roots of a Real Cubic Equation.    It suffices to consider

f(x)=x^3 -3lx+q                                  (l$$0),

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

($$l,     q-2l$$l),    (-$$l,     q+2l$$l),

which are shown by crosses in Figs. 18-20 for the graph of y=f(x) in the

q $$ 2l$$l                       q $$ -2l$$l
FIG. 18                          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.

-2l$$l $$ q $$ 2l$$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
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$$0,
and a triple root if q^2 = 4l^3 = 0. 
