Question

Verify Lagrange’s mean value theorem for the function \( f(x) = x + 4 \) on the interval [0, 5].

Hint:

MVT requires continuity and differentiability; for linear functions, the derivative is constant.

Answer 1

Lagrange’s MVT: If \( f(x) \) is continuous on [a, b] and differentiable on (a, b), there exists \( c \in (a, b) \) such that: \[ f'(c) = \frac{f(b) - f(a)}{b - a}. \] For \( f(x) = x + 4 \), [0, 5]:
- Continuous and differentiable everywhere.
- \( f(0) = 4 \), \( f(5) = 5 + 4 = 9 \).
\[ \frac{f(5) - f(0)}{5 - 0} = \frac{9 - 4}{5} = 1. \] \[ f'(x) = 1 \quad \forall x. \] For any \( c \in (0, 5) \), \( f'(c) = 1 \), which equals \( \frac{f(5) - f(0)}{5} \).
MVT holds for all \( c \in (0, 5) \).