Question: Prove that the equation has only one real root.
Proof:
The existence of at least one real root follows from the fact that the above polynomial is an odd power.
Let us prove the uniqueness of such a root by contradiction.
Suppose there exist two roots . Then, in the interval
the given polynomial function satisfies all the conditions of Rolle’s Theorem: it is continuous, it vanishes at the end points and has derivative at all points. Consequently, at some point
such that
,
equals zero. But,
. But, this contradicts hypothesis. Hence, the proof.
More later,
Nalin Pithwa