Q224 / 270 · NDA 2016 (Phase 2) · Mathematics

Which one of the following statements is correct in respect of the function $f(x)=x^3 \sin x$ ?

  • A
    It has local maximum at $x=0$.
  • Correct answer
    B
    It has local minimum at $\mathrm{x}=0$.
  • C
    It has neither maximum nor minimum at $x=0$.
  • D
    It has maximum value 1.

Solution

$\begin{aligned} & f(x)=x^3 \sin x \\ & f^{\prime}(x)=3 x^2 \sin x+x^3 \cos x \\ & f^{\prime}(x)=0 \\ & \Rightarrow 3 x^2 \sin x+x^3 \cos x=0 \\ & \Rightarrow x^2(3 \sin x+x \cos x)=0 \\ & \Rightarrow x=0,3 \sin x+x \cos x=0 \\ & \text { Put } x=0 \text { in }(1) \\ & 3 \sin x=0 \Rightarrow \sin x=0 \\ & f^{\prime \prime}(x)=6 x \sin x+3 x^2 \cos x+3 x^2 \cos x+x^3(-\sin x) \\ & f^{\prime \prime}(0)=0 \end{aligned}$ So, $\mathrm{f}(\mathrm{x}) \min$. at $x=0$.

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

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