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JEE Main202126 Feb 2021Morning ShiftMathematicsContinuity and DifferentiabilityActual

Let f be any function defined on R and let it satisfy the condition: f x - f y ≤ x - y 2 , ∀ x , y ∈ R . If f 0 = 1 , then :

Options

  1. Af x = 0 , ∀ x ∈ R
  2. Bf x can take any value in R
  3. Cf x < 0 , ∀ x ∈ R
  4. Df x > 0 , ∀ x ∈ R

Correct answer

D. f x > 0 , ∀ x ∈ R

Step-by-step solution

f x - f y x - y ≤ x - y Let x - y = h ⇒ x = y + h lim x → 0 f y + h - f y h ≤ 0 ⇒ f ' y ≤ 0 ⇒ f ' y = 0 ⇒ f y = k (constant) And f 0 = 1 given So, f y = 1 ⇒ f x = 1

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