Show that the equationx5 − 6x + c = 0
where c is a constant has at most one real root on the interval [−1, 1].
Assume to the contrary that there are more than one root of the equation in the interval . That is, there are at least two roots in this interval. Pick any two roots, if there are only two, pick both. Let the roots be and , with (without loss of generality). Then we have . Thus, by Rolle's theorem, there exists an in the interval such that . Now . But there is no real for which this attains zero in much less . Thus we have a contradiction. Therefore, has at most one root in