JEE Advanced2015MathematicsContinuity and DifferentiabilityActual
Let g : R → R be a differentiable function with g 0 = 0 , g ′ 0 = 0 and g ′ 1 ≠ 0 . Let f x = x | x | g x , x ≠ 0 0 , x = 0 and h x = e | x | for all x ∈ R . Let foh ( x ) denote f h x a n d ( hof ) ( x ) denote h f x .Then which of the following is (are) true?
Options
- Af is differentiable at x = 0
- Bh is differentiable at x = 0
- Cf ₀ h is differentiable at x = 0
- Dh ₀ f is differentiable at x = 0
Correct answer
A. f is differentiable at x = 0
Step-by-step solution
f x = g x x > 0 0 x = 0 - g x x < 0 L H D : lim x → 0 - - g ′ x = - lim x → 0 - g ′ x = - 0 = 0 R H D : lim x → 0 + g ′ x = g ′ 0 + = 0 Hence f x is differentiable at x = 0 h x = e x = e x x ≥ 0 e - x x < 0 L H D : lim x → 0 - - e - x = - 1 R H D : lim x → 0 + e x = 1 Hence h x is not differentiable at x = 0 f h x = e x e x g e x f h x = g e x x ≥ 0 g e - x x < 0 L H D : lim x → 0 + g ′ e - x × - e - x = g ′ 1 × - 1 = - g ′ 1 R H D : lim x → 0 + g ′ e x