Verify Rolle's theroem

Dear Student, As f (x) = (x - 1) {(x - 2)}^{2} = (x - 1) (x^{2} - 4 x + 4) = x^{3} - 5 x^{2} + 8 x - 4 since it is a polynomial function so it is differentiable and continuous everywhere . Conditions : (1) f (x) is continuous on [1, 2] (2) f (x) is differentiable on (1, 2) and f' (x) = 3 x^{2} - 10 x + 8 also, f (1) = 0 = f (2) Hence condition for roll' s theorem is satisfied, then there must exist atleast one value of c \in (1, 2) such that f' (c) = 0 3 c^{2} - 10 c + 8 = 0 3 c^{2} - 6 c - 4 c + 8 = 0 3 c (c - 2) - 4 (c - 2) = 0 (c - 2) (3 c - 4) = 0 c = 2, \frac{4}{3} but in the interval [1, 2] c = \frac{4}{3} is acceptable so c = \frac{4}{3} Hence rolles theorem is verified Regards

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