$Let f (x) = \sin^{- 1} (x) + \cos^{- 1} (x^{2}) + \sin^{- 1} (x^{3}) + \dots . . + \sin^{- 1} (x^{2 n - 1}) + \cos^{- 1} (x^{2 n}) and$ $g (x) = \cos^{- 1} x + \sin^{- 1} (x^{2}) + \cos^{- 1} (x^{3}) + .... + \cos^{- 1} (x^{2 n - 1}) + \sin^{- 1} (x^{2 n})$ $w h e r e n i s a p o s i t i v e i n t e g e r . I f m i n i m u m o f (f (x)) + m a x i m u m o f g (x) = 8 π, t h e n n =$

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answer is B.

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Detailed Solution

$min of f (x) and maximum of g (x) occur at x = - 1 m i n o f f (x) + m a x i m u m o f g (x) a t x = - 1 i s$

$= f (- 1) + g (- 1) = \frac{π}{2} + \frac{π}{2} + \frac{π}{2} . . . . . + \frac{π}{2} (2 n t i m e s) = 8 π (\sin c e \sin^{- 1} x + \cos^{- 1} x = \frac{π}{2}) \Rightarrow n π = 8 π \Rightarrow n = 8$

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