Analysis - Prove that there exists a smallest positive number $p$ such that $\cos(p) = 0$ 
Prove that there exists a smallest positive number $p$ such that $\cos(p)  = 0
$.

I think I'm supposed to use either Rolle's theorem or the Mean Value Theorem but I'm not sure how, any help would be appreciated. Thanks!

How about you restrict the domain to I=(0,pi) and show that since cos(x) is monotonic on I, it is one to one, and therefore only one value, p=pi/2, would be the solution on I since cos(a)=cos(b)=0 --> a=b, the solution p in I is unique and since any other positive solution "q" would have to be outside the interval I, pi/2=p
