Question

In the interval \( (0, 1) \), the function \( f(x) = x^2 - x + 1 \) is

• A. Strictly decreasing
• B. Increasing
• C. Neither increasing nor decreasing
• D. Decreasing

Hint:

To determine if a function is increasing or decreasing, check the sign of the derivative. If the derivative is positive, the function is increasing; if negative, it is decreasing. If the derivative changes sign, the function is neither increasing nor decreasing.

Answer 1

The Correct Option is C

Step 1: Find the derivative of \( f(x) \).
To determine if the function is increasing or decreasing, we first find the derivative of the function \( f(x) = x^2 - x + 1 \). Using basic differentiation rules:
\[ f'(x) = \frac{d}{dx} (x^2 - x + 1) = 2x - 1 \]

Step 2: Analyze the sign of the derivative.
The derivative \( f'(x) = 2x - 1 \) represents the rate of change of the function. To determine the behavior of \( f(x) \) in the interval \( (0, 1) \), we examine the sign of \( f'(x) \) within this interval.
- For \( x = 0 \), \( f'(0) = 2(0) - 1 = -1 \) (negative).
- For \( x = 1 \), \( f'(1) = 2(1) - 1 = 1 \) (positive).

Step 3: Conclusion.
Since \( f'(x) = 2x - 1 \) changes sign from negative to positive as \( x \) moves from 0 to 1, the function is neither strictly increasing nor strictly decreasing in the interval \( (0, 1) \).
Thus, the correct answer is option (C): Neither increasing nor decreasing.