In which of the following functions Rolle's theorem is applicable?

1) f (x) = |x| in – 2 ≤ x ≤ 2

2) f (x) = tanx in 0 ≤ x ≤ π

3) f (x) = 1 + (x – 2) in 1 ≤ x ≤ 3

4) f (x) = x (x – 2)2 in 0 ≤ x ≤ 2

Solution: (4) f (x) = x (x – 2)2 in 0 ≤ x ≤ 2

(a) not differentiable at x = 0 ∈ [-2, 2]

(b) At x = π / 2, tanx is discontinuous

(c) At x = 1, it has complex roots

(d) f (x) = x (x – 2)2 – it is continuous and differentiable

f (0) = f (2) = 0 – Rolle’s theorem

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