Verify lagranges mean value theorem for the function `f(x)=(x-3)(x-6)(x-9) on [3, 5]`

Explanation

To verify Lagrange's Mean Value Theorem (LMVT) for the function \( f(x) = (x-3)(x-6)(x-9) \) on the interval \([3, 5]\), we will follow these steps:

### Step 1: Check the conditions of LMVT
Lagrange's Mean Value Theorem states that if a function \( f(x) \) is continuous on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), then there exists at least one \( c \) in \((a, b)\) such that:
\[ f'(c) = \frac{f(b) - f(a)}{b - a} \]
Here, \( a = 3 \) and \( b = 5 \).
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