Question

Assertion (A): The function \( f(x) = x^2 - x + 1 \) is strictly increasing on \((-1, 1)\). Reason (R): If \( f(x) \) is continuous on \([a, b]\) and derivable on \((a, b)\), then \( f(x) \) is strictly increasing on \([a, b]\) if \( f'(x)>0 \) for all \( x \in (a, b) \).

Hint:

A function is strictly increasing if \( f'(x)>0 \) for all \( x \) in the given interval. If \( f'(x) \) changes sign, the function is not strictly increasing.

Answer 1

Step 1: Compute the derivative of \( f(x) = x^2 - x + 1 \): \[ f'(x) = 2x - 1. \] 
Step 2: Analyze the sign of \( f'(x) \) on \((-1, 1)\): - At \( x = \frac{1}{2} \), \( f'(x) = 0 \). - For \( x<\frac{1}{2} \), \( f'(x)<0 \), meaning \( f(x) \) is decreasing. - For \( x>\frac{1}{2} \), \( f'(x)>0 \), meaning \( f(x) \) is increasing. 
Step 3: Since \( f(x) \) is not strictly increasing throughout \((-1,1)\), Assertion (A) is false. 
Step 4: Reason (R) states a correct mathematical theorem, so it is true. Thus, the correct answer is that Assertion (A) is false, but Reason (R) is true.