Try this beautiful problem on Sign change, useful for ISI B.Stat Entrance.

## Sign change | ISI B.Stat Entrance | Problem 709

In the interval \((-2\pi, 0)\) the function \(f(x) = sin(1/x^3)\)

- (a) never changes sign
- (b) changes sign only once
- (c) changes sign more than once, but a finite number of times
- (d) changes sign infinite number of times

**Key Concepts**

Calculus

Limit

Trigonometry

## Check the Answer

But try the problem first…

Answer: (d)

TOMATO, Problem 709

Challenges and Thrills in Pre College Mathematics

## Try with Hints

First hint

As x becomes \(\leq 1\) and tends to zero then it crosses \(\pi, 2\pi, 3\pi, ….\).can you draw the graph?

Can you now finish the problem ……….

Final Step

If we draw the graph then we can see that the function \(f(x) = sin(1/x^3)\) crosses many times. Therefore number of sign changes is infinite.

Therefore option \((d)\) is correct…..

## Other useful links

- https://www.cheenta.com/cubical-box-amc-10a-2010-problem-20/
- https://www.youtube.com/watch?v=kYVpq7qzxpA