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

Correct Answer: 248

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(4^x – 2^x)
∵ f(2) = 3 then
\(k = \frac{1}{4}\)
\(∴ ƒ(x) = \frac{4x - 2x}{4}\)
\(∴ ƒ′(x) = \frac{4^xIn4 - 2^xIn2}{4}\)
\(ƒ′(x) = \frac{(2.4^x - 2^x ) In2}{4}\)
∴ \(\frac{ƒ′(4)}{ƒ′(2)}= \frac{2.256 - 16}{2.16 - 4}\)
\(∴ 14 \frac{ƒ′(4)}{ƒ′(2)} = 248\)