JEE Advanced2016MathematicsDifferential EquationsActual
Let f : 0 , ∞ → R be a differentiable function such that f ' x = 2 - f x x ∀ x ∈ ( 0 , ∞ ) and f 1 ≠ 1 . Then,
Options
- Alim x → 0 + ⁡ f ′ 1 x = 1
- Blim x → 0 + ⁡ x f 1 x = 2
- Clim x → 0 + ⁡ x 2 f ′ x = 0
- Df x ≤ 2 for all x ∈ 0 ,   2
Correct answer
A. lim x → 0 + ⁡ f ′ 1 x = 1
Step-by-step solution
Let y = f x d y d x + y x = 2 (linear first order differential equation) I . F . = e ∫ d x x = e log e x = x , (I.F.=Integrating factor) ∴   y · x = 2 ∫ x d x + c ∴   y x = x 2 + c ⇒ f x = x + c x ; As f 1 ≠ 1   ⇒ c ≠ 0 ⇒ f ' x = 1 - c x 2 ,   c   ≠ 0 lim x → 0 + ⁡ f ' 1 x = lim x → 0 + ⁡ 1 - c x 2 = 1 lim x → 0 + ⁡ x f 1 x = lim x → 0 + ⁡ x 1 x + c x = lim x → 0 +