To determine the interval where the function \( f(x) = x^2 - 4x + 6 \) is increasing, we follow these steps.
1. Calculate the First Derivative:
The first derivative of \( f(x) \) is found by:
\( f'(x) = \frac{d}{dx}(x^2 - 4x + 6) = 2x - 4 \)
2. Find Critical Points:
Set the first derivative to zero to find critical points:
\( 2x - 4 = 0 \Rightarrow x = 2 \)
This critical point divides the number line into two intervals: \( (-\infty, 2) \) and \( (2, \infty) \).
3. Analyze the Sign of the Derivative:
- For \( x<2 \), let \( x = 1 \). \( f'(1) = 2(1) - 4 = -2 \). The derivative is negative, indicating the function is decreasing.
- For \( x>2 \), let \( x = 3 \). \( f'(3) = 2(3) - 4 = 2 \). The derivative is positive, indicating the function is increasing.
4. Conclusion:
The function is strictly increasing in the interval \( (2, \infty) \). While \( x = 2 \) is a stationary point where the derivative is zero, the interval of increase begins immediately after \( x = 2 \). Given the options, the closest representation of the increasing interval is:
(D) [2, ∞)
Final Answer:
The function is increasing in the interval [2, ∞).