Given f(x) = x3 – 3x2 + 2x + 5
As we know that, every polynomial function is continuous as well as differentiable.
So, f(x) is continuous and differentiable on the indicated interval.
Also, f(0) = 5 and f(2) = 8 – 12 + 4 + 5 = 5
i.e. f(0) = 5 = f(2)
Thus, all the conditions of Rolle’* theorem are satisfied.
Now, we have to show that there exist a point c in (0, 2) such that f'(c) = 0
We have f'(c) = 3c2 – 6c + 2 = 0 = 0 gives
We have f'(c) = 3c2 – 6c + 2 = 0 gives
⇒ c = (6 ± √(36 – 24))/6 = (6 ± 2√3)/6
= 1 ± 1/√3
⇒ c = 1 ±1/√3 ∈ (0, 2)
Hence, Rolle’* theorem is verified.
