Question

Let $P(x)$ be a polynomial of degree 2, with $P(2) = -1$, $P'(2) = 0$, $P''(2) = 2$, then $P(1.001)$ is

• A. -0.001999
• B. -0.002
• C. -1
• D. 0.001999

Hint:

When a polynomial's value and derivatives at a specific point are given, consider using the Taylor expansion around that point. This often simplifies the polynomial's form and subsequent calculations.

Answer 1

The Correct Option is A

Step 1: Determine the general form of a polynomial of degree 2. A polynomial of degree 2 can be written as $P(x) = ax^2 + bx + c$.
Step 2: Find the first and second derivatives of $P(x)$. The first derivative is $P'(x) = 2ax + b$. The second derivative is $P''(x) = 2a$.
Step 3: Use the given conditions to find the coefficients $a, b, c$. Given $P''(2) = 2$: \[ 2a = 2 \implies a = 1 \] Now, $P'(x) = 2(1)x + b = 2x + b$. Given $P'(2) = 0$: \[ 2(2) + b = 0 \implies 4 + b = 0 \implies b = -4 \] Now, $P(x) = (1)x^2 + (-4)x + c = x^2 - 4x + c$. Given $P(2) = -1$: \[ (2)^2 - 4(2) + c = -1 \] \[ 4 - 8 + c = -1 \] \[ -4 + c = -1 \] \[ c = 3 \] So, the polynomial is $P(x) = x^2 - 4x + 3$. Alternative Step 3 (Using Taylor expansion around $x=2$): A polynomial can be expressed using its Taylor series expansion around a point $x_0$: \[ P(x) = P(x_0) + P'(x_0)(x-x_0) + \frac{P''(x_0)}{2!}(x-x_0)^2 + ....... \] Since $P(x)$ is a degree 2 polynomial, higher order derivatives are zero. Given $x_0 = 2$, $P(2) = -1$, $P'(2) = 0$, $P''(2) = 2$. \[ P(x) = P(2) + P'(2)(x-2) + \frac{P''(2)}{2}(x-2)^2 \] \[ P(x) = -1 + 0(x-2) + \frac{2}{2}(x-2)^2 \] \[ P(x) = -1 + (x-2)^2 \] This simplified form of $P(x)$ is equivalent to $x^2 - 4x + 4 - 1 = x^2 - 4x + 3$.
Step 4: Evaluate $P(1.001)$. Using the simplified form $P(x) = (x-2)^2 - 1$: \[ P(1.001) = (1.001 - 2)^2 - 1 \] \[ P(1.001) = (-0.999)^2 - 1 \] \[ P(1.001) = (0.999)^2 - 1 \] We can write $0.999$ as $(1 - 0.001)$. \[ P(1.001) = (1 - 0.001)^2 - 1 \] Expand $(1 - 0.001)^2$: \[ (1 - 0.001)^2 = 1^2 - 2(1)(0.001) + (0.001)^2 \] \[ = 1 - 0.002 + 0.000001 \] Substitute this back: \[ P(1.001) = (1 - 0.002 + 0.000001) - 1 \] \[ P(1.001) = -0.002 + 0.000001 \] \[ P(1.001) = -0.001999 \]