# Math Help - Taylor problem #1

1. ## Taylor problem #1

Let n>=2 and p(x)=x^n + a1*x^(n-1) + ... + an .
Prove that to equation p(x)=sin(x) have at most n roots.

2. Originally Posted by Also sprach Zarathustra
Let n>=2 and p(x)=x^n + a1*x^(n-1) + ... + an .
Prove that to equation p(x)=sin(x) have at most n roots.
Between each two zeros of $f(x) = p(x)-\sin x$ there will be a zero of $f'(x)$. So $f(x)$ can have at most one more zero than $f'(x)$. Then between each two zeros of $f'(x)$ there will be a zero of $f''(x)$, and so on.

Now look at what happens when you differentiate $f(x)$ n times.