Periodic Function | TOMATO B.Stat Objective 710

If f(x) = \(a_0+a_1cosx+a_2cos2x+....+a_ncosnx\) where \(a_0,a_1,....,a_n\) are non zero real numbers and \(a_n > |a_0|+|a_1|+....+|a_{n-1}|\), then number of roots of f(x)=0 in \( 0 \leq x \leq 2\pi\), is

f is periodic with period \(2\pi\)

here \(0 <|\displaystyle\sum_{k=0}^{n-1}a_kcos(kx)| \leq \displaystyle\sum_{k=0}^{n-1}|a_k|<a_n, x\in\)set of reals

for points \(x_k=\frac{k\pi}{n}\) \(1 \leq k \leq 2n\)

\(f(x_k)=a_n[cos(k\pi)+\theta], |\theta|<1 for 1 \leq k \leq 2n\)

[ since cos \(\theta\) is periodic and f(x) is expressed for every point x=\(x_k\)]

f has at least 2n points in such a period interval where f has alternating sign.

I.S.I. & C.M.I. Entrance Program

Taught by students and alumni of I.S.I. & C.M.I.

Learn More

Subscribe to Cheenta at Youtube