162 CHAPTER 3 Inequalities
The D = 0 case is the one we
shall find to have many applica-
tions, especially to constrained ex-
trema problems. Indeed, assuming
this to be the case, and denotingα
as the double root off(x) = 0, then
we havef(x) =a(x−α)^2 and that
the graph of y = f(x) appears as
depicted to the right.
y 6xy=f(x)x=αThe geometrical implication of the double real root is that the graph
ofy=f(x) not only has anx-intercept at x=α, it is tangent to the
x-axis at this point. We shall find this observation extremely useful, as
it provides application to a wealth of constrained extrema problems.
The first example is rather pedestrian, but it will serve to introduce
the relevant methodology.
Example 1. Find the minimum value of the quadratic function
f(x) = 2a^2 − 12 x+ 23.Solution. If we denote this minimum
bym, then the graph ofy=mwill
be tangent to the graph ofy=f(x)
where this minimum occurs. This
says that in solving the quadratic
equationf(x) =m, there must be
a double root, i.e., the discriminant
of the quadratic 2x^2 − 12 x+ 23−m
must vanish. The geometry of this
situation is depicted to the right. -y 6xy =f(x)