Question

If the parabola y = f(x), having axis parallel to y-axis, touches the line y = x at (1, 1) then

• A. 2f′(0) + f(0) = 1
• B. 2f(0) + f′(0) = 1
• C. 2f(0) – f′(0) = 1
• D. 2f′(0) – f(0) = 1

Answer 1

Answer: (B)

2f(0) + f′(0) = 1

Answer 2

Let f(x) = ax² + bx + c

⇒ f′ (x) = 2ax + b

Thus, f(0) = C and f′(0) = b

We have that information that y = f(x) is being touched by

y = x at (1, 1)

⇒ ax² + bx + c = x

should have x = 1 at its repeated root.

⇒ ax² + x(b –1) + c = 0 are 1, 1

⇒ $\frac{1 - b}{a}$= 2, $\frac{c}{a}$= 1

⇒ 1 – b = 2a = 2c

⇒ 1 – f′ (0) = 2f(0)

or 2f (0) + f′(0) = 1