JEE Mains · Maths · STD 12 - 6. Application of derivatives
Let \(f(\mathrm{x})=\mathrm{x} \cos ^{-1}(-\sin |\mathrm{x}|), \quad \mathrm{x} \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right],\) then which of the following is true?
- A \(f^{\prime}\) is decreasing in \(\left(-\frac{\pi}{2}, 0\right)\) and increasing \(\operatorname{in}\left(0, \frac{\pi}{2}\right)\)
- B \(f\) is not differentiable at \(x=0\)
- C \(f^{\prime}(0)=-\frac{\pi}{2}\)
- D \(f^{\prime}\) is increasing in \(\left(-\frac{\pi}{2}, 0\right)\) and decreasing \(\operatorname{in}\left(0, \frac{\pi}{2}\right)\)
Answer & Solution
Correct Answer
(A) \(f^{\prime}\) is decreasing in \(\left(-\frac{\pi}{2}, 0\right)\) and increasing \(\operatorname{in}\left(0, \frac{\pi}{2}\right)\)
Step-by-step Solution
Detailed explanation
\(f(\mathrm{x})\) is an odd function. Now, if \(x \geq 0,\) then \(f(x)=x \cos ^{-1}(-\sin x)\) \(=x\left(\frac{\pi}{2}-\sin ^{-1}(-\sin x)\right)=x\left(\frac{\pi}{2}+x\right)\) Hence,…
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