I am having a lot of trouble with this problem, any help would be greatly appreciated!
Prove that the that the trigonometric polynomial $$a_0 + a_1\cos(x)+\cdots+a_n\cos(nx), $$ where the coefficients are all real and $|a_0|+|a_1|+\cdots+|a_{n-1}|\leq a_n$, has at least $2n$ zeros in the interval $[0,2\pi)$.
I am pretty sure I have to use the Intermediate Value Theorem at some point but I am not really sure how to use it properly in this situation. Thank you for any help!
Evaluate your formula at $k\pi / n$ where $k = 0,1,2,\ldots,2n$, and focus on the leading term vs the sum of the other terms. You will get that the formula is negative for odd $k$ and positive for even $k$, based on the inequality on the coefficients and the fact that $|\cos x|$ is strictly less than 1 when $x$ is not a multiple of $\pi$. Then apply the intermediate value theorem like you said.