Verify Lagranges mean value theorem for function `f(x)=x^3-5x^2-3x` on `[1,\ 3]`

Explanation

To verify Lagrange's Mean Value Theorem for the function \( f(x) = x^3 - 5x^2 - 3x \) on the interval \([1, 3]\), we will follow these steps:

### Step 1: Check the conditions of the Mean Value Theorem
1. **Continuity**: The function \( f(x) \) is a polynomial function, and all polynomial functions are continuous everywhere. Thus, \( f(x) \) is continuous on \([1, 3]\).
2. **Differentiability**: The function \( f(x) \) is also differentiable everywhere since it is a polynomial. Therefore, it is differentiable on \((1, 3)\).

### Step 2: Calculate \( f(a) \) and \( f(b) \)
- Let \( a = 1 \) and \( b = 3 \).
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