Mathematics — Real Analysis · rolle-original-theorem
Problem
Prove the theorem Rolle actually proved in 1691: If f is a polynomial, f′(a)=f′(b)=0 for some a<b, and there is no c∈(a,b) such that f′(c)=0, then there is at most one root of f in (a,b), that is at most one x∈(a,b) such that f(x)=0. In other words, between any two consecutive roots of f′ is at most one root of f. Hint: Suppose there are two roots and see what happens.