The discussion focuses on demonstrating that the function cos(cos x) is a contraction mapping using the Mean Value Theorem (MVT). The derivative of the function, |d/dx (cos(cos x))|, is expressed as |sin(cos x) sin(x)|, which is shown to be less than sin(1), confirming the contraction property. Participants seek clarification on how to establish that |sin(cos x) sin(x)| is indeed less than sin(1).
PREREQUISITESMathematics students, educators, and anyone interested in advanced calculus, particularly those studying contraction mappings and the Mean Value Theorem.
zbot1 said:Homework Statement
how to show using MVT that cos(cos x) is a contraction.
Homework Equations
| d/dx (cos(cos x)) | = | sin(cos x) sin(x) | < sin 1 < 1
The Attempt at a Solution
Using that relation, the original problem is easily solved. My question is, how do we know:
| sin(cos x) sin(x) | < sin 1 ?
-zbot1