Question

The function \( f(x) = x^2 - 4x + 6 \) is increasing in the interval:

• A. \( (0, 2) \)
• B. \( (-\infty, 2] \)
• C. \( [1, 2] \)
• D. \( [2, \infty) \)

Hint:

To determine where a function is increasing or decreasing, find the first derivative and solve the inequality \( f'(x)>0 \) for increasing or \( f'(x)<0 \) for decreasing.

Answer 1

The Correct Option is D

To solve the problem, we need to find the interval where the function \( f(x) = x^2 - 4x + 6 \) is increasing.

1. Take the First Derivative:
To determine where a function is increasing, we compute its first derivative:

\( f'(x) = \frac{d}{dx}(x^2 - 4x + 6) = 2x - 4 \)

2. Set Derivative Equal to Zero to Find Critical Point:
Set \( f'(x) = 0 \):
\( 2x - 4 = 0 \Rightarrow x = 2 \)
This divides the real number line into two intervals: \( (-\infty, 2) \) and \( (2, \infty) \)

3. Test the Sign of the Derivative:
- For \( x < 2 \): Choose \( x = 1 \), then \( f'(1) = 2(1) - 4 = -2 \) → Negative → Function is decreasing.
- For \( x > 2 \): Choose \( x = 3 \), then \( f'(3) = 2(3) - 4 = 2 \) → Positive → Function is increasing.

4. Conclusion:
The function is increasing in the interval \( (2, \infty) \). Since option (D) is \( [2, \infty) \), and derivative at \( x = 2 \) is 0, which is a stationary point, the function is not increasing at exactly \( x = 2 \). So the strictly increasing interval is \( (2, \infty) \), but the closest matching answer is:
(D) [2, ∞)

Final Answer:
The function is increasing in the interval [2, ∞).