- #1

issacnewton

- 1,026

- 36

## Homework Statement

Consider the function ##f(x) = x^4 \sin(\frac 1 x)## for ##x \ne 0## and ##f(x) = 0## for ##x =0##. I have to prove that ##x=0## is the critical number of this function and its derivative changes the sign indefinitely.

## Homework Equations

Definition of the critical number

## The Attempt at a Solution

##x=0## would be the critical number if ##f'(0) = 0## or if ##f'(0)## does not exist. I have been able to show that ##f'(0) = 0## using the squeeze theorem. Now I want to show that the derivative of ##f(x)## changes sign indefinitely. I am totally stuck at this point. How would I progress from here ?

Thanks ##\smallsmile##