Question

The function \( f(x) = x^2 - x + 1 \) is:

• A. Increasing in \( (0, 1) \)
• B. Decreasing in \( (0, 1) \)
• C. Increasing in \( (0, \frac{1}{2}) \) and decreasing in \( (\frac{1}{2}, 1) \)
• D. Increasing in \( (\frac{1}{2}, 1) \) and decreasing in \( (0, \frac{1}{2}) \)

Hint:

For increasing or decreasing intervals, solve \( f'(x) = 0 \) and test the sign of \( f'(x) \) in the resulting intervals.

Answer 1

The Correct Option is D

Step 1: Differentiate \( f(x) = x^2 - x + 1 \): \[ f'(x) = 2x - 1. \] Step 2: Solve \( f'(x) = 0 \): \[ 2x - 1 = 0 \quad \Rightarrow \quad x = \frac{1}{2}. \] Step 3: Analyze the sign of \( f'(x) \): - \( f'(x)<0 \) in \( (0, \frac{1}{2}) \), meaning \( f(x) \) is decreasing. - \( f'(x)>0 \) in \( (\frac{1}{2}, 1) \), meaning \( f(x) \) is increasing. Thus, the function is decreasing in \( (0, \frac{1}{2}) \) and increasing in \( (\frac{1}{2}, 1) \).