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Check whether the conditions of Rolle’s theorem are satisfied by the function f (x) = (x - 1) (x - 2) (x - 3), x ∈ [1, 3] - Mathematics and Statistics

Check whether the conditions of Rolle’s theorem are satisfied by the function
f (x) = (x - 1) (x - 2) (x - 3), x ∈ [1, 3]

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Solution

`f(x)=(x-1)(x-2)(x-3),`          `x in[1, 3]`

`=x^3-6x^2+11x-6`

As f(x) is a polynomial function, it is continuous and differentiable everywhere on its domain. Thus,
a. f(x) is continuous on [1, 3]
b. f(x) is differentiable on (1, 3)

Further, f(1) = 0 and f(3) = 0
∴ f(1) = f(3)

Thus, all the conditions of Rolle’s theorem are satisfied.

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