Q93 / 155 · WBJEE 2023 · Mathematics

In the interval \((-2 \pi, 0)\), the function \(f(x)=\sin \left(\frac{1}{x^3}\right)\)

  • A
    never changes sign
  • B
    changes sign only once
  • C
    changes sign more than once but finitely many times
  • Correct answer
    D
    changes sign infinitely many times

Solution

Hint : \(x \in(-2 \pi, 0), f(x)=\sin \left(\frac{1}{x^3}\right)\) \(\begin{aligned} & \because-2 \pi < x < 0 \\ & \Rightarrow-8 \pi^3 < x^3 < 0 \\ & \Rightarrow-\infty < \frac{1}{x^3} < -\frac{1}{8 \pi^3} \end{aligned}\) Hence, \(\sin \left(\frac{1}{x^3}\right)\) will take all values from -1 to 1 , and will change its sign infinitely many times since it is a periodic function

Hence, the correct answer is option: 'D'.

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