How do I show that a derivative of a polynomial has a zero in an interval?

[Charles49]

Homework Statement

If an even degree polynomial of order 2n intersects the x-axis twice, how do I show that the (2n-1) th derivative has a zero in that interval?

Homework Equations

example: let g(x)=x^3(1-x). Show without computation that g'''(c) =0 for some c in (0, 1).

The Attempt at a Solution

I know that g' has a zero at x=1/2 because that is a maximum. I don't know how I can be sure that the 2nd and 3rd derivative have a zero in this interval.

 

[lanedance]

consider what happens to terms less of order (2n-2) and less...