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

Step 1: Finding the first derivative of \( f(x) \).
The function given is \( f(x) = x^2 - 4x + 6 \). To find the interval where the function is increasing, we first calculate its derivative: \[ f'(x) = 2x - 4 \] Step 2: Determining when the derivative is positive.
The function is increasing where \( f'(x)>0 \). Thus, solve for \( x \) in: \[ 2x - 4>0 \quad \Rightarrow \quad x>2 \] Therefore, the function \( f(x) \) is increasing for \( x \in [2, \infty) \).