Mean Value Theorem Proof
The Mean Value Theorem is a fundamental concept in calculus that establishes a relationship between the average rate of change of a function over an interval and the instantaneous rate of change at a specific point within that interval.
Statement of the Mean Value Theorem: Let f(x) be a continuous function on the closed interval [a, b] and differentiable on the open interval (a, b). Then, there exists a number c in (a, b) such that: \[ f'(c) = \frac{f(b) - f(a)}{b - a} \]
Proof of the Mean Value Theorem: Consider the function g(x) = f(x) - \[ \left( \frac{f(b) - f(a)}{b - a} \right) \times (x - a) \] This function is continuous on [a, b] and differentiable on (a, b).
Now, let's define another function h(x) = g(x) / (x - a), x \(\neq\) a. By the definition of h(x), we have: h(x) = \[ \frac{f(x) - f(a)}{x - a} - \frac{f(b) - f(a)}{b - a} \]
Since h(x) is continuous on (a, b) and differentiable on (a, b), by Rolle's Theorem, there exists a number c in (a, b) such that h'(c) = 0. Therefore, we have: \[ \frac{f'(c) \times (c - a) - (f(b) - f(a))}{c - a} = 0 \]
Solving for f'(c) gives us: \[ f'(c) = \frac{f(b) - f(a)}{b - a} \] This concludes the proof of the Mean Value Theorem.