Question

Let f: R→R and f be a differentiable function such that f(x + 2y) = f(x) + 4f(y) + 2y(2x – 1) ∀ x, y ∈ R and f’(0) = 1, then f(3) + f’(3) is ______.

Options

• 16.00

• 17.00

• 18.00

• 19.00

Answer 1

Let f: R→R and f be a differentiable function such that f(x + 2y) = f(x) + 4f(y) + 2y(2x – 1) ∀ x, y ∈ R and f’(0) = 1, then f(3) + f’(3) is 19.00.

Explanation:

Put x = y = 0

⇒ f(0) = 0

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

= `lim_(h→0) (f(x) + 4f(h/2) + h(2x - 1) - f(x))/h`

f’(x) = `lim_(h→0) (4f(h/2) - 4f(0) + h(2x - 1))/h`

f’(x) = 2f’(0) + (2x – 1)

⇒ f’(x) = 2x + 1

⇒ f(x) = x² + x

Now, f(3) + f’(3) = 12 + 7 = 19