3. The expression \( \sqrt{-x^{2}+6 x-5} \) has a (A) maximum value of 2 (B) minimum value of 2 (C) maximum value of 3 (D) minimum value of 3

To determine the maximum or minimum value of the expression \( \sqrt{-x^{2} + 6x - 5} \), follow these steps: 1. **Determine the Domain:** The expression inside the square root must be non-negative: \[ -x^{2} + 6x - 5 \geq 0 \] This can be rewritten as: \[ x^{2} - 6x + 5 \leq 0 \] Factoring the quadratic equation \( x^{2} - 6x + 5 = 0 \) gives: \[ (x - 1)(x - 5) = 0 \implies x = 1 \text{ or } x = 5 \] Therefore, the domain is \( 1 \leq x \leq 5 \). 2. **Find the Maximum and Minimum of the Quadratic Expression:** Consider the quadratic function \( y = -x^{2} + 6x - 5 \). - This is a downward-opening parabola (since the coefficient of \( x^2 \) is negative). - The vertex (maximum point) occurs at: \[ x = \frac{-b}{2a} = \frac{-6}{2(-1)} = 3 \] - Substituting \( x = 3 \) into \( y \): \[ y = -(3)^2 + 6(3) - 5 = -9 + 18 - 5 = 4 \] - Thus, the maximum value of \( y \) is 4. 3. **Determine the Range of the Square Root:** - The square root function \( \sqrt{y} \) will range from \( \sqrt{0} = 0 \) to \( \sqrt{4} = 2 \). 4. **Conclusion:** The expression \( \sqrt{-x^{2} + 6x - 5} \) attains a **maximum value of 2** within its domain. **Answer:** **(A) maximum value of 2**