# Real Analysis : Differentiable and Increasing Functions

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.

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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. "ïƒ¨" 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

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#### Solution Summary

Differentiable and Increasing are investigated.