Rolle’s Theorem states that, Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b), such that f(a) = f(b), where a and b are some real numbers.Then there exists some c in (a, b) such that f’(c) = 0.
We have, f(x) = |x – 1| in [0, 2].
Since, polynomial and modulus functions are continuous everywhere f(x) is continuous.
We need to check if f(x) is differentiable at x=1 or not.
= (∵ f(x) = 1-x, if x< 1)
= (∵ f(x) = x-1, if 1< x)
⇒ Lf’(1) ≠ Rf’(1)
⇒ f(x) is not differentiable at x=1.
Hence, rolle’s theorem is not applicable on f(x) since it is not differentiable at x=1 ∈ (0,2).
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