Let f (x) = (x^2 − 1)^n . Prove (by induction on r) that for r = 0, 1, 2, · · · , n,(adsbygoogle = window.adsbygoogle || []).push({});

f^ (r) (x)(the r-th derivative of f(x)) is a polynomial whose value is 0 at no fewer than r distinct points of (−1, 1).

I'm thinking about expanding f(x) as the sum of the (n+1) terms, then it's easier to take derivatives. But I don't know how to get the roots from there then. Could anyone please give me some hints? Thanks!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Roots of higher derivatives

**Physics Forums | Science Articles, Homework Help, Discussion**