Given function is f (x) = x (x – 4)2 on [0, 4]
Since, given function f is a polynomial it is continuous and differentiable everywhere i.e., on R.
Let us find the values at extremes:
⇒ f (0) = 0(0 – 4)2
⇒ f (0) = 0
⇒ f (4) = 4(4 – 4)2
⇒ f (4) = 4(0)2
⇒ f (4) = 0
∴ f (0) = f (4), Rolle’* theorem applicable for function ‘f’ on [0,4].
Let’* find the derivative of f(x):
f'(x) = d(x(x - 4)2)/dx
Differentiate using product rule.
Here, f'(c) = 0, c ∈ (0,4), from the definition of rolle'* theorem
Thus, Rolle'* theorem is verified.
