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JEE Advanced2016MathematicsDifferentiationActual

Let f : R → R , g : R → R and h : R → R be differentiable functions such that f x = x 3 + 3 x + 2 , g f x = x and h g g x = x , for all x ∈ R . Then,

Options

  1. Ag ′ 2 = 1 15
  2. Bh ′ 1 = 666
  3. Ch 0 = 16
  4. Dh g 3 = 36

Correct answer

B. h ′ 1 = 666

Step-by-step solution

If f ′ x = 3 x 2 + 3 ⇒ g f x = x ⇒ g ′ f x = 1 f ′ x Also f 0 = 2  ( Put x = 0 ) So, g ′ 2 = 1 f ′ 0 ⇒ g ′ 2 = 1 3 If h ( g g x = x h ′ g g x = 1 g ′ g x . g ′ x If, g g x = 1 g ( f x ) = x ⇒ g - 1 x = f ( x ) ⇒ g x = g - 1 1 ⇒ g x = f 1 = 6 To find, g ' 6 use x = 1 in g ' f x = 1 f ' 1 as   f 1 = 6 To find, g ' 236 , use x = 6 as f 6 = 236 ∴ h ' 1 = 1 g ' 6 . g ' ( 236 ) = 1 1 6 . 1

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