Check whether the conditions of Rolle's theorem are satisfied by the function \(f(x) = x^2 - 4x + 3,\ x \in [1, 3]\)
Q.11 Check whether the conditions of Rolle's theorem are satisfied by the function \(f(x) = x^2 - 4x + 3,\ x \in [1, 3]\)
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Answer: Yes, all three conditions are satisfied (and \(c = 2\)).
(1) \(f(x)\) is a polynomial, so it is continuous on \([1, 3]\).
(2) \(f'(x) = 2x - 4\) exists for all \(x\), so \(f\) is differentiable on \((1, 3)\).
(3) \(f(1) = 1 - 4 + 3 = 0\) and \(f(3) = 9 - 12 + 3 = 0\), so \(f(1) = f(3)\).
Hence the conditions of Rolle's theorem are satisfied. Indeed \(f'(c) = 0 \Rightarrow 2c - 4 = 0 \Rightarrow c = 2 \in (1, 3)\).
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· 28 Oct 2026
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