Verify Rolle's theorem for the function f(x) = `x^(3) + 3x^(2) - 24x -80` in the interval `[-4,5]`.

Explanation

To verify Rolle's theorem for the function \( f(x) = x^3 + 3x^2 - 24x - 80 \) in the interval \([-4, 5]\), we will follow these steps:

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

1. **Continuity**: Since \( f(x) \) is a polynomial function, it is continuous everywhere, including the interval \([-4, 5]\).
2. **Differentiability**: Polynomial functions are also differentiable everywhere, so \( f(x) \) is differentiable on the interval \((-4, 5)\).
3. **Check \( f(-4) \) and \( f(5) \)**:
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