To find the minimum or maximum of a quadratic we
complete the square expressing the function in the form
If
the
minimum will be where
so
and
the minimum is at
the
minimum will be wheresoand
the minimum is at
If
the
maximum will be wheresoand
the maximum is at
the
maximum will be wheresoand
the maximum is at
For example, to find the minimum of
complete the square to get
then
the minimum is at
To find the maximum of
complete
the square to get
then the maximum is at
complete
the square to get
then the maximum is at
We might also have to find the maxima of reciprocal
quadratics such as
The quadratic here can have no roots if it is to have a
maximum, or else at those roots we would have
which
has no value, and close to those roots the graph would tend to
As
before we complete the square to get
To
maximise y we have to minimise the denominator ie minimise
This
has a minimum at
hence
has
a maximum at
This
is illustrated below. If the numerator were negative we would follow
the same procedure, completing the square but now find a minimum, in
this case at
which
has no value, and close to those roots the graph would tend to
As
before we complete the square to getTo
maximise y we have to minimise the denominator ie minimiseThis
has a minimum athencehas
a maximum atThis
is illustrated below. If the numerator were negative we would follow
the same procedure, completing the square but now find a minimum, in
this case at
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