Q.23 Verify Rolle's theorem for the function f(x) = x² - 5x + 9,\ x in [1, 4]

Answer: Rolle's theorem is verified with c = 5/2 in (1, 4) . f(x) is a polynomial, so it is continuous on [1, 4] and differentiable on (1, 4) . f(1) = 1 - 5 + 9 = 5 and f(4) = 16 - 20 + 9 = 5 , so f(1) = f(4) . All conditions hold, so there should be c in (1, 4) with f'(c) = 0 . f'(x) = 2x - 5 ; f'(c) = 0 ⇒ c = 5/2 = 2.5 , which lies in (1, 4) . Hence Rolle's theorem is verified.

Verify Rolle's theorem for the function \(f(x) = x^2 - 5x + 9,\ x \in [1, 4]\)

Q.23 Verify Rolle's theorem for the function \(f(x) = x^2 - 5x + 9,\ x \in [1, 4]\)

Doubtly Asked 28 Oct 2026 Board exam · June 2026 hsc 12th Save WhatsApp Report

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Answer: Rolle's theorem is verified with \(c = \frac{5}{2} \in (1, 4)\).

\(f(x)\) is a polynomial, so it is continuous on \([1, 4]\) and differentiable on \((1, 4)\).
\(f(1) = 1 - 5 + 9 = 5\) and \(f(4) = 16 - 20 + 9 = 5\), so \(f(1) = f(4)\).

All conditions hold, so there should be \(c \in (1, 4)\) with \(f'(c) = 0\).

\(f'(x) = 2x - 5\); \(f'(c) = 0 \Rightarrow c = \frac{5}{2} = 2.5\), which lies in \((1, 4)\). Hence Rolle's theorem is verified.

Doubtly AI · 28 Oct 2026 AI-verified · Oct 2026 Report

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Verify Rolle's theorem for the function \(f(x) = x^2 - 5x + 9,\ x \in [1, 4]\)

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