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
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
on the interval and
on the interval ,
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.