Question

Let f: R→R be a polynomial function satisfying f(x + y) = f(x) + f(y) + 3xy(x + y) –1 ∀ x, y ∈ R and f'(0) = 1, then `lim_(x→∞)(f(2x))/(f(x)` is equal to ______.

Options

• 6.00

• 7.00

• 8.00

• 9.00

Answer 1

Let f: R→R be a polynomial function satisfying f(x + y) = f(x) + f(y) + 3xy(x + y) –1 ∀ x, y ∈ R and f'(0) = 1, then `lim_(x→∞)(f(2x))/(f(x)` is equal to 8.00.

Explanation:

f'(x) = `lim_(h→0)(f(x + h) - f(x))/h`

= `lim_(h→0)(f(h) + 3hx^2 + 3xh^2 - 1)/h`

∵ At x = y = 0 in functional rule: f(0) = 1

∴ f’(x) = f’(0) + 3x² + 0

⇒ f’(x) = 3x² + 1

⇒ f(x) = x³ + x + C

∵ f(0) = 1

⇒ f(x) = x³ + x + 1

`lim_(x→∞) (f(2x))/(f(x))` = 8