Verify Rolles theorem for function `f(x)=x(x-2)^2` on `[0,\ 2]`

Explanation

To verify Rolle's theorem for the function \( f(x) = x(x-2)^2 \) on the interval \([0, 2]\), we will follow these steps:

### Step 1: Check Continuity and Differentiability
Rolle's theorem states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point \( c \) in the open interval such that \( f'(c) = 0 \).

1. **Continuity**: The function \( f(x) = x(x-2)^2 \) is a polynomial function. Polynomial functions are continuous everywhere, including the closed interval \([0, 2]\).

2. **Differentiability**: Since \( f(x) \) is a polynomial, it is also differentiable everywhere, including the open interval \((0, 2)\).
...