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Question: An extremum value of the function \(f(x) = \left( \sin^{- 1}x \right)^{3} + \left( \cos^{- 1}x \rig...

An extremum value of the function

f(x)=(sin⁡−1x)3+(cos⁡−1x)3(−1<x<1)f(x) = \left( \sin^{- 1}x \right)^{3} + \left( \cos^{- 1}x \right)^{3}( - 1 < x < 1) is

A

7π38\frac{7\pi^{3}}{8}

B

π38\frac{\pi^{3}}{8}

C

π332\frac{\pi^{3}}{32}

D

π316\frac{\pi^{3}}{16}

Answer

π332\frac{\pi^{3}}{32}

Explanation

Solution

f′(x)=3(sin⁡−1x)21−x2−3(sin⁡−1x)21−x2f'(x) = \frac{3\left( \sin^{- 1}x \right)^{2}}{\sqrt{1 - x^{2}}} - \frac{3\left( \sin^{- 1}x \right)^{2}}{\sqrt{1 - x^{2}}}

For maxima & minima

f′(x)=0f'(x) = 0

⇒ 3[(sin⁡−1x)2−(cos⁡−1x)2]1−x2\frac{3\left\lbrack \left( \sin^{- 1}x \right)^{2} - \left( \cos^{- 1}x \right)^{2} \right\rbrack}{\sqrt{1 - x^{2}}}

⇒ (sin⁡−1x)2−(cos⁡−1x)2=0\left( \sin^{- 1}x \right)^{2} - \left( \cos^{- 1}x \right)^{2} = 0

⇒ sin⁡−1x=cos⁡−1x\sin^{- 1}x = \cos^{- 1}x ⇒ x = 12\frac{1}{\sqrt{2}}

Extremum value of f(x)f(x) is

f(12)=π364+π364=π332f\left( \frac{1}{\sqrt{2}} \right) = \frac{\pi^{3}}{64} + \frac{\pi^{3}}{64} = \frac{\pi^{3}}{32}

So. 'C' is correct.