Verify Lagrange's mean theorem for the function `f(x) =x^(2) + x-1 ` in the interval [0,4]

Explanation

To verify Lagrange's Mean Theorem for the function \( f(x) = x^2 + x - 1 \) in the interval \([0, 4]\), we will follow these steps:

### Step 1: Check the continuity of the function
The function \( f(x) = x^2 + x - 1 \) is a polynomial function. Polynomial functions are continuous everywhere. Therefore, \( f(x) \) is continuous on the closed interval \([0, 4]\).

### Step 2: Check the differentiability of the function
Since \( f(x) \) is a polynomial function, it is also differentiable everywhere. Thus, \( f(x) \) is differentiable on the open interval \((0, 4)\).

...