Question

If \( x \) is real and \( \alpha, \beta \) are maximum and minimum values of \( \frac{x^2 - x + 1}{x^2 + x + 1} \) respectively, then \( \alpha + \beta = \):

• A. \( \frac{10}{3} \)
• B. \( \frac{8}{3} \)
• C. \( \frac{4}{3} \)
• D. \( \frac{-2}{3} \)

Hint:

For rational functions, sometimes analyzing the behavior as \( x \to \infty \) or solving the derivative can provide insights into the maximum and minimum values.

Answer 1

The Correct Option is A

Step 1: Expression for the function. We are given the function: \[ f(x) = \frac{x^2 - x + 1}{x^2 + x + 1}. \] Step 2: Differentiating the function. We differentiate the function with respect to \(x\) using the quotient rule: \[ f'(x) = \frac{(2x - 1)(x^2 + x + 1) - (x^2 - x + 1)(2x + 1)}{(x^2 + x + 1)^2}. \] Step 3: Solving for the critical points. We solve for the critical points by setting the numerator of \(f'(x)\) equal to zero. 
Step 4: Evaluating the maximum and minimum values. After evaluating the function at the critical points, we find the maximum and minimum values of \(f(x)\) to be \( \frac{10}{3} \) . 
Step 5: Sum of the maximum and minimum values. The sum is: \[ \frac{10}{3} \]