Question

The function \( f(x) = x^2 - 6x + 8 \) is positive when

• A. \( x<2 \)
• B. \( 2<x<4 \)
• C. \( -6 \leq x \leq 8 \)
• D. \( x>4 \)

Hint:

For a quadratic function, the roots indicate the points where the function changes sign. The parabola opens upwards when the coefficient of \( x^2 \) is positive.

Answer 1

The Correct Option is A, D

Step 1: Solving the inequality.
The function \( f(x) = x^2 - 6x + 8 \) is a quadratic function, and we need to find where it is positive. First, solve \( f(x) = 0 \): \[ x^2 - 6x + 8 = 0 \] Factoring gives: \[ (x - 2)(x - 4) = 0 \] Thus, \( x = 2 \) and \( x = 4 \) are the roots of the equation.
Step 2: Analyzing the function.
Since the parabola opens upwards (positive coefficient for \( x^2 \)), the function is positive when \( x>4 \) or \( x<2 \). The correct answer is therefore (D).