Step 1: Calculate the derivative of \( f(x) = x^2 - x + 1 \): \( f'(x) = 2x - 1 \). Step 2: Find the critical points by setting \( f'(x) = 0 \): \( 2x - 1 = 0 \), which yields \( x = \frac{1}{2} \). Step 3: Examine the sign of \( f'(x) \) in the intervals \( (0, \frac{1}{2}) \) and \( (\frac{1}{2}, 1) \). For \( x \in (0, \frac{1}{2}) \), \( f'(x)<0 \), indicating \( f(x) \) is decreasing. For \( x \in (\frac{1}{2}, 1) \), \( f'(x)>0 \), indicating \( f(x) \) is increasing. Therefore, \( f(x) \) decreases on \( (0, \frac{1}{2}) \) and increases on \( (\frac{1}{2}, 1) \).