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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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) $.

cdquestions Admin · Accepted Answer

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.

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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