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Question: An extreme value of the function f(x) = (sin<sup>–1</sup>x)<sup>3</sup> + (cos<sup>–1</sup> x)<sup>...

An extreme value of the function

f(x) = (sin–1x)3 + (cos–1 x)3 (–1 < x < 1) is

A

π316\frac{\pi^{3}}{16}

B

π332\frac{\pi^{3}}{32}

C

π38\frac{\pi^{3}}{8}

D

7π38\frac{7\pi^{3}}{8}

Answer

π332\frac{\pi^{3}}{32}

Explanation

Solution

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

for extremum f ¢(x) = 0

Ž (sin–1(x))2 – (cos–1x)2=0

Ž (sin–1x – cos–1x) (sin–1x + cos–1x) = 0

Ž sin–1 x – cos–1 x = 0 Q sin–1x + cos–1x = p/2}

Ž x = 12\frac{1}{\sqrt{2}}

\ extreme valuef(12)f\left( \frac{1}{\sqrt{2}} \right)

= (sin⁡−112)3\left( \sin^{- 1}\frac{1}{\sqrt{2}} \right)^{3} +(cos⁡−112)3\left( \cos^{- 1}\frac{1}{\sqrt{2}} \right)^{3}

= (p/4)3 + (p/4)3 = π332\frac{\pi^{3}}{32}