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 \( x \in (a, b) \).

Question

Let \( \mathbb{R}_+ \) denote the set of all non-negative real numbers. Then the function \( f : \mathbb{R}_+ \to \mathbb{R}_+ \) defined as \( f(x) = x^2 + 1 \) is:

Answer 1

Step 1: Calculate the derivative of \( f(x) = x^2 - x + 1 \). The derivative is \( f'(x) = 2x - 1 \).
Step 2: Examine the sign of \( f'(x) \) on the interval \((-1, 1)\).
- When \( x = \frac{1}{2} \), \( f'(x) = 0 \).
- When \( x<\frac{1}{2} \), \( f'(x)<0 \), indicating \( f(x) \) is decreasing.
- When \( x>\frac{1}{2} \), \( f'(x)>0 \), indicating \( f(x) \) is increasing.
Step 3: Assertion (A) is false because \( f(x) \) is not strictly increasing over the entire interval \((-1,1)\).
Step 4: Reason (R) is true as it states a valid mathematical theorem. Therefore, Assertion (A) is false, and Reason (R) is true.

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 \( x \in (a, b) \).

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