Question

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 ___.

Answer 1

The correct answer is 248
∵ f(x + y) = 2 x f(y) + 4 y f(x) …(1)
Now, f(y + x) 2 y f(x) + 4 x f(y) …(2)
∴ 2 x f(y) + 4 y f(x) = 2 y f(x) + 4 x f(y)
(4 y – 2 y) f(x) = (4 x – 2 x) f(y)
$\frac{ƒ(x)}{4x - 2x} = \frac{ƒ(y)}{4y - 2y} = k (Say)$
∴ f(x) = k(4x – 2x)
∵ f(2) = 3 then
$k = \frac{1}{4}$
$∴ ƒ(x) = \frac{4x - 2x}{4}$
$∴ ƒ′(x) = \frac{4^xIn4 - 2^xIn2}{4}$
$ƒ′(x) = \frac{(2.4^x - 2^x ) In2}{4}$
∴ $\frac{ƒ′(4)}{ƒ′(2)}= \frac{2.256 - 16}{2.16 - 4}$
$∴ 14 \frac{ƒ′(4)}{ƒ′(2)} = 248$