Explore BrainMass

# Real Analysis : Differentiable and Increasing Functions

Not what you're looking for? Search our solutions OR ask your own Custom question.

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

A-a function f:(a,b)->R is increasing on (a,b) if f(x)<=f(y) whenever x<y in (a,b). Assume f is differentiable on (a,b). Show that f is increasing on (a,b)if and only if f'(x)>=0 for all x belong to (a,b).
b-show that the function g(x){x/(2+x^2 sin(1/x)) if x not=0 0 if x=0
is differentiable on R and satisfies g'(0)>0.Now prove that g is not increasing over any open interval containing 0.

© BrainMass Inc. brainmass.com September 26, 2022, 8:16 am ad1c9bdddf
https://brainmass.com/math/real-analysis/real-analysis-differentiable-increasing-functions-30042

#### Solution Preview

Please see the attached file for the complete solution.
Thanks for using BrainMass.

A) A function f:(a,b)->R is increasing on (a,b) if f(x)<=f(y) whenever x<y in (a,b). Assume f is differentiable on (a,b). show that f is increasing on (a,b)if and only if f'(x)>=0 for all x belong to (a,b).
Proof. "" If f is increasing on (a,b), then f'(x)>=0 for all x belong to (a,b).
We can prove it by contradiction. Assume that there exists a point such that . Since f is differentiable on (a,b), f is differentiable at . BY definition, we have
...

#### Solution Summary

Differentiable and Increasing are investigated.

\$2.49