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JEE Advanced2014MathematicsApplication of DerivativesActual

For every pair of continuous functions f , g : 0 , 1 → R such that max ⁡ f x : x ∈ 0 , 1 = max ⁡ g x : x ∈ 0 , 1 , then the correct statement (s) is (are)

Options

  1. Af c 2 + 3 f c = g c 2 + 3 g c for some c ∈ 0 , 1
  2. Bf c 2 + f c = g c 2 + 3 g c for some c ∈ 0 , 1
  3. Cf c 2 + 3 f c = g c 2 + g c for some c ∈ 0 , 1
  4. Df c 2 = g c 2 for some c ∈ 0 , 1

Correct answer

A. f c 2 + 3 f c = g c 2 + 3 g c for some c ∈ 0 , 1

Step-by-step solution

Let f x and g x achieve their maximum value at x 1 and x 2 respectively h x = f x - g x h x 1 = f x 1 - g x 1 ≥ 0 h x 2 = f x 2 - g x 2 ≤ 0 ⇒ h c = 0 where c ∈ 0 , 1 ⇒ f c = g c .

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