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JEE Main202226 Jun 2022Evening ShiftMathematicsDifferentiationActual

Let f : ℝ → ℝ satisfy f x + y = 2 x f y + 4 y ( f x , ∀ x , y ∈ ℝ . If f 2 = 3 , then 14 · f ' 4 f ' 2 is equal to _____.

Correct answer

0

Step-by-step solution

Given, f x + y = 2 x f y + 4 y ( f x Put y = 2 we get, f x + 2 = 2 x × 3 + 16 f x f ' x + 2 = 16 f ' x + 3 × 2 x ln 2 Now put x = 2 we get, f ' 4 = 16 f ' 2 + 12 ln 2           … i Similarly, f y + 2 = 4 f y + 3 × 4 y f ' 4 = 4 f ' y + 3 × 4 y ln 4 f ' 4 = 4 f ' 2 + 96 ln 2           ⋯ ii solving eq. i and ii , we get f ' 2 = 7 ln 2 from equation i , we get f ' 4 = 124 ln 2 Now ⇒ 14 × f ' 4 f ' 2 14

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