If f (x )= sin ( cos^-1 ( 1-2^2x1+2^2x )and its first derivative with respect to x is - ba log_e2 when x = 1, where a and b are integers, then the minimum value of (a^2-b^2)is ____.

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Accepted Answer

Correct answer: 481. (481) f (x )= sin ( cos^-1 ( 1-2^2x1+2^2x ) ) atx=1 ;2^2x=4 for sin ( cos^-1 ( 1-2^2x1+2^2x ) ) let tan^-1x= ; ∈(- 2 , 2) ∴ sin ( cos^-1 cos2 )= sin2 ( Ifx>1 ⇒ 2> > 4 ∴ >2 > 2) =2 sin cos = 2 tan 1+ tan^2 ∴f^1 (1 )= 20 ln2-32 ln225=- 1225 ln2 So, a=25 ,b=12 ⇒(a^2-b^2)=25^2-12^2 =625-144 =481

$If f (x) = sin (cos^{- 1} (\frac{1 - 2 ^{2 x}}{1 + 2 ^{2 x}}) and its first derivative with respect to x is - \frac{b}{a} log_{e} 2 when x = 1, where a and b are integers, then the minimum value of (a^{2} - b^{2}) is ____.$

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