If f x = sin cos - 1 1 - 2 2 x 1 + 2 2 x and its first derivative with respect to x is - b a log e 2 when x = 1 , where a and b are integers, then the minimum value of a 2 - b 2 is _______.

Step-by-step solution

Given f x = sin cos - 1 1 - 2 2 x 1 + 2 2 x  at  x = 1 ; 2 2 x = 2 x 2 For sin cos - 1 1 - 2 x 2 1 + 2 x 2 Let tan - 1 2 x = θ ; θ ∈ - π 2 , π 2 ∴ sin cos - 1 cos 2 θ = sin 2 θ  If  x > 1 ⇒ π 2 > θ > π 4 ∴   π > 2 θ > π 2 = 2 sin θ cos θ = 2 tan θ 1 + tan 2 θ Hence, f x = 2 · 2 x 1 + 2 2 x ∴ f ' x = 1 + 2 2 x 2 . 2 x ln 2 - 2 2 x · 2 · ln 2 &#1