Given as the function is f(x) = x(x – 2)2 on [0, 2]
Here, given function f is a polynomial it is continuous and differentiable everywhere that is on R.
To find the values at extremes:
⇒ f (0) = 0(0 – 2)2
⇒ f (0) = 0
⇒ f (2) = 2(2 – 2)2
⇒ f (2) = 2(0)2
⇒ f (2) = 0
f (0) = f(2), Rolle’* theorem applicable for function f on [0,2].
Let us find the derivative of f(x)
f'(x) = d(x(x - 2)2)/dx
Differentiate using UV rule.
f'(c) = 0, c ∈ (0,1), from the definition of rolle'* theorem
Thus, Rolle'* theorem is verified.
