Verify Lagrange's mean value theorem for the function ` f(x) = 2x^(2) -10 x + 29` in the interval [2.7]

Explanation

To verify Lagrange's Mean Value Theorem (LMVT) for the function \( f(x) = 2x^2 - 10x + 29 \) in the interval \([2, 7]\), we will follow these steps:

### Step 1: Check Continuity and Differentiability
Lagrange's Mean Value Theorem states that if a function 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:
\[ f'(c) = \frac{f(b) - f(a)}{b - a} \]
The function \( f(x) = 2x^2 - 10x + 29 \) is a polynomial, which is continuous and differentiable everywhere. Therefore, it is continuous on \([2, 7]\) and differentiable on \((2, 7)\).
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