Expert: Abe Mantell - 6/2/2006

Question
how to prove that equation cos x = x has at least one root using intermediate value theorem

Answer
Hello,

The IVT states:
If f is continuous on a closed interval [a,b], and c is any number between f(a) and f(b) inclusive, then there is at least one number x in the closed interval such that f(x)=c.

So, we write cos(x)=x as cos(x)-x=0 and show that
f(x)=cos(x)-x has at least one root.
f(0)=1 and f(pi/2)=-pi/2, since c=0 is between f(0) and f(pi/2),
then there exists at least one number x (a root) between
0 and pi/2 such that f(x)=0 -- by the IVT.

Abe