`"Verify LMVT for " f(x) =-x^(2)+4x-5 " and " x in [-1,1]`

Explanation

To verify the Lagrange Mean Value Theorem (LMVT) for the function \( f(x) = -x^2 + 4x - 5 \) on the interval \([-1, 1]\), we will follow these steps:

### Step 1: Check the conditions of LMVT
The function \( f(x) \) must be continuous on the closed interval \([-1, 1]\) and differentiable on the open interval \((-1, 1)\).

**Solution:**
The function \( f(x) = -x^2 + 4x - 5 \) is a polynomial, and all polynomial functions are continuous and differentiable everywhere. Therefore, \( f(x) \) is continuous on \([-1, 1]\) and differentiable on \((-1, 1)\).

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