Question

Find the intervals in which the function \( f(x) = x^2 - 4x + 6 \) is 

Hint:

To determine intervals of increase or decrease, find the derivative, set it equal to zero to find critical points, and test intervals around those points.

Answer 1

Step 1: The first derivative of the function \( f(x) \) will give us information about where the function is increasing or decreasing: \[ f'(x) = \frac{d}{dx}(x^2 - 4x + 6) = 2x - 4. \] Step 2: Set \( f'(x) = 0 \) to find the critical points: \[ 2x - 4 = 0 \quad \Rightarrow \quad x = 2. \] Step 3: Now, we analyze the sign of \( f'(x) \) on intervals determined by the critical point \( x = 2 \). - For \( x<2 \), \( f'(x) = 2x - 4<0 \), so the function is decreasing on \( (-\infty, 2) \).
- For \( x>2 \), \( f'(x) = 2x - 4>0 \), so the function is increasing on \( (2, \infty) \). Thus, the function is decreasing on \( (-\infty, 2) \) and increasing on \( (2, \infty) \). \bigskip