Suppose . In this problem, we will show that has
exactly one root (or zero) in the interval .
(a) First, we show that has a root in the interval .
Since is a
function on the
interval and
and
,
the graph of must cross the -axis
at some point in the interval by the
.
Thus, has at least one root in the interval
.
(b) Second, we show that cannot have more than one
root in the interval by a thought experiment.
Suppose that there were two roots and
in the interval with . Then
.
Since is
on the interval and
on the interval ,
by
there would exist a point in interval
so that .
However, the only solution to is
, which is not
in the interval , since .
Thus, cannot have more than one root in .
(Note: where the problem asks you to make a choice select the weakest choice that works in the given context. For example "continuous" is a weaker condition than "polynomial" because every polynomial is continuous but not vice-versa. Rolle's theorem is a weaker theorem than the mean value theorem because Rolle's theorem applies to fewer cases.)
You can earn partial credit on this problem.