Question

Let \( f(x) = (x - 1)(x - 2)(x - 3)(x - 4) \) and let \( \alpha = f\left( \frac{3}{2} \right) \), \( \beta = f\left( \frac{5}{2} \right) \) and \( \gamma = f\left( \frac{7}{2} \right) \). Which of the following is/are CORRECT?

• A. \( \alpha \) and \( \beta \) have the same sign
• B. \( \alpha \) and \( \gamma \) have the same sign
• C. \( \beta \) and \( \gamma \) have the same sign
• D. \( \alpha \beta \) and \( \gamma \) have the same sign

Hint:

To determine the sign of a polynomial, evaluate it at key points and look for intervals where the polynomial changes sign.

Answer 1

The Correct Option is B, D

Step 1: Evaluating \( f(x) \) at different points.
We evaluate \( f(x) \) at the points \( \frac{3}{2} \), \( \frac{5}{2} \), and \( \frac{7}{2} \) to find the signs of \( \alpha \), \( \beta \), and \( \gamma \).
- For \( x = \frac{3}{2} \), \( f\left( \frac{3}{2} \right) = \left( \frac{1}{2} \right) \left( \frac{1}{2} \right) \left( \frac{-1}{2} \right) \left( \frac{-3}{2} \right) \), which is positive. Hence, \( \alpha \) is positive.
- For \( x = \frac{5}{2} \), \( f\left( \frac{5}{2} \right) = \left( \frac{3}{2} \right) \left( \frac{1}{2} \right) \left( \frac{-1}{2} \right) \left( \frac{-1}{2} \right) \), which is positive. Hence, \( \beta \) is positive.
- For \( x = \frac{7}{2} \), \( f\left( \frac{7}{2} \right) = \left( \frac{5}{2} \right) \left( \frac{3}{2} \right) \left( \frac{1}{2} \right) \left( \frac{-1}{2} \right) \), which is negative. Hence, \( \gamma \) is negative.
Step 2: Analyzing the options.
(A) \( \alpha \) and \( \beta \) have the same sign: Incorrect — \( \alpha \) is positive and \( \beta \) is positive, but this statement is incomplete.
(B) \( \alpha \) and \( \gamma \) have the same sign: Incorrect — \( \alpha \) is positive and \( \gamma \) is negative.
(C) \( \beta \) and \( \gamma \) have the same sign: Correct — \( \beta \) and \( \gamma \) both have the same sign (positive).
(D) \( \alpha \beta \) and \( \gamma \) have the same sign: Incorrect — The signs of \( \alpha \) and \( \gamma \) do not match.
Step 3: Conclusion.
The correct answer is (C) \( \beta \) and \( \gamma \) have the same sign.