Prove that if an/(n+1) + an-1/n + ... + a1/2 + a0 = 0 (ak in R), then the equation anx^n + an-1x^(n-1) + ... + a1x + a0 = 0 has at least one real root between 0 and 1.
This is my work.
Let p(x) = (an/n+1)x^(n+1) + ..... a0x.
Then p'(x) = anx^n + ......... + a0 (precisely what we need to show has a root). This part I'm not sure about. If p(x) has two distinct real roots (call them a and b and WLOG b < d), then p(b) = p(d) = 0. Rolle's Theorem on the interval [b,d] tells us that there exists a number r such that p'(r) = 0, which is the root I'm looking for. Does this make sense? Then, all I'm unsure about is how to show p(x) has two distinct real roots and how the interval [0,1] plays in the problem.