Verify Rolle’s theorem for each of the following functions on the indicated intervals :
f(x) = (x – 1) (x – 2)2 on [1, 2]
First let us write the conditions for the applicability of Rolle’s theorem:
For a Real valued function ‘f’:
a) The function ‘f’ needs to be continuous in the closed interval [a,b].
b) The function ‘f’ needs differentiable on the open interval (a,b).
c) f(a) = f(b)
Then there exists at least one c in the open interval (a,b) such that f’(c) = 0.
Given function is:
⇒ f(x) = (x – 1)(x – 2)2 on [1,2]
Since, given function f is a polynomial it is continuous and differentiable everywhere i.e, on R.
Let us find the values at extremums:
⇒ f(1) = (1 – 1)(1 – 2)2
⇒ f(1) = 0(1)2
⇒ f(1) = 0
⇒ f(2) = (2 – 1)(2 – 2)2
⇒ f(2) = 1.02
⇒ f(2) = 0
∴ f(1) = f(2), Rolle’s theorem applicable for function ‘f’ on [1,2].
Let’s find the derivative of f(x):
⇒
Differentiating using UV rule
⇒
⇒ f’(x) = ((x – 2)2×1) + ((x – 1)×2×(x – 2))
⇒ f’(x) = x2 – 4x + 4 + 2(x2 – 3x + 2)
⇒ f’(x) = 3x2 – 10x + 8
We have f’(c) = 0 cϵ(1,2), from the definition given above.
⇒ f’(c) = 0
⇒ 3c2 – 10c + 8 = 0
⇒
⇒
⇒
⇒
⇒ ϵ(1,2) (neglecting the value 2)
∴ Rolle’s theorem is verified.