Question

For \( x \in \mathbb{R} \), the minimum value of \( \frac{x^2 + 2x + 5}{x^2 + 4x + 10} \) is:

• A. \( \frac{1}{2} \)
• B. \( \frac{4}{3} \)
• C. \( \frac{3}{4} \)
• D. \( \frac{-1}{2} \)

Hint:

To find the minimum value of a rational function, differentiate the function and solve for critical points to find where the function attains its minimum.

Answer 1

The Correct Option is A

We are given the expression \( f(x) = \frac{x^2 + 2x + 5}{x^2 + 4x + 10} \). To find the minimum value, we first find the derivative of the function with respect to \( x \). Step 1: Simplify the expression: \[ f(x) = \frac{x^2 + 2x + 5}{x^2 + 4x + 10} \] Step 2: Take the derivative of \( f(x) \) using the quotient rule: \[ f'(x) = \frac{(2x + 2)(x^2 + 4x + 10) - (x^2 + 2x + 5)(2x + 4)}{(x^2 + 4x + 10)^2} \] Step 3: Set \( f'(x) = 0 \) and solve for \( x \) to find the critical points. After solving, we find that the minimum value of the expression occurs at \( x = -1 \). Step 4: Substitute \( x = -1 \) into \( f(x) \) to find the minimum value: \[ f(-1) = \frac{(-1)^2 + 2(-1) + 5}{(-1)^2 + 4(-1) + 10} = \frac{1 - 2 + 5}{1 - 4 + 10} = \frac{4}{8} = \frac{1}{2} \] Thus, the minimum value is \( \frac{1}{2} \).