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Let f : R → R be a function such that f x + y 3 = f x + f y 3 , f 0 = 0 and f ′ 0 = 5 , then

Options

  1. Af x is a quadratic function
  2. Bf x is continuous but not differentiable
  3. Cf x is differentiable in R
  4. Df x is bounded in R

Correct answer

C. f x is differentiable in R

Step-by-step solution

We have, f x + y 3 = f x + f y 3 , f 0 = 0 and f ′ 0 = 5 f ′ x = lim h → 0 f x + h − f x h = lim h → 0 ⁡ f 3 x + 3 h 3 - f 3 x + 0 3 h = l i m h → 0 f 3 x + f 3 h 3 - f 3 x + f 0 3 h = lim h → 0 f 3 h − f 0 3 h = f ′ 0 = 5 ∴ f x = 5 x + c ∴ f 0 = 0 ⇒ c = 0 Hence, f x = 5 x

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