Question

y = f(x) is a parabola, having it's axis parallel to y-axis. If the line y = x touches this parabola at x = 1, then

• A. f”(1) – f’ (0) = 1
• B. f”(0) – f’(1) = 1
• C. f”(1) + f’(0) = 1
• D. f”(0) + f’(1) = 1

Answer 1

Answer: (C)

f”(1) + f’(0) = 1

Answer 2

Let y = f(x) = ax² + bx + c. We have f(1) = 1

⇒ a + b + c = 1. Also ax² + bx + c = x should have x – 1 as it's repeated root.

⇒ ax² + (b – 1)x + c = a(x – 1)²

⇒ 1 – b = 2a, a = c.

We have f '(x) = 2ax + b, f'(x) = 2a

⇒ f(1) = 2a, f(0) = b

⇒ f "(1) + f(O) = 1.