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)

Source
Suggested Reading

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

Sign change - graph

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…..

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