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JEE Main20199 Jan 2019Evening ShiftMathematicsContinuity and DifferentiabilityActual

Let f be a differentiable function from R to R such that f x - f y ≤ 2 x - y 3 / 2 , for all x , y ∈ R . If f 0 = 1 then ∫ 0 1 f 2 x d x is equal to

Options

  1. A0
  2. B1
  3. C2
  4. D1 2

Correct answer

B. 1

Step-by-step solution

Given functional equality is f x - f y ≤ 2   x - y 3 / 2   Put x = y + h ,   we get f y + h - f y ≤ 2 .   | h 3 / 2 ⇒ f y + h - f ( y ) h ≤ 2   h 1 / 2 ⇒ lim h → 0 ⁡ f y + h - f y h ≤ lim h → 0 ⁡ 2   h 1 / 2 ⇒ f ′ y ≤ 0 ⇒ f ′ y = 0 ⇒ f ′ y = 0 ⇒ f y = c ⇒ f y = 1   (since f 0 = 1 ) Now ∫ 0 1 f 2 x d x =   ∫ 0 1 1   d x = 1 .

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