Suppose [math] . In this problem, we will show that [math] has
exactly one root (or zero) in the interval [math] .
(a) First, we show that [math] has a root in the interval [math] .
Since [math] is a
choose
continuous
differentiable
polynomial
function on the
interval [math] and
[math]
and
[math]
,
the graph of [math] must cross the [math] -axis
at some point in the interval [math] by the
choose
intermediate value theorem
mean value theorem
squeeze theorem
Rolle's theorem
.
Thus, [math] has at least one root in the interval
[math] .
(b) Second, we show that [math] cannot have more than one
root in the interval [math] by a thought experiment.
Suppose that there were two roots [math] and
[math] in the interval [math] with [math] . Then
[math]
.
Since [math] is
choose
continuous
differentiable
polynomial
on the interval [math] and
choose
continuous
differentiable
polynomial
on the interval [math] ,
by
choose
intermediate value theorem
mean value theorem
squeeze theorem
Rolle's theorem
there would exist a point [math] in interval [math]
so that [math] .
However, the only solution to [math] is
[math]
, which is not
in the interval [math] , since [math] .
Thus, [math] cannot have more than one root in [math] .

(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.)