Mean Value Theorem

The Mean Value Theorem is a fundamental theorem in calculus that provides a connection between the concepts of continuity and differentiability . It...

The Mean Value Theorem is a fundamental theorem in calculus that provides a connection between the concepts of continuity and differentiability. It states that for any function defined on an open interval within a closed interval, there exists a number (c) in that interval such that the average rate of change of the function over that interval is equal to the instantaneous rate of change at (c).

That is,

$f'(c) = \frac{f(b) - f(a)}{b - a}$

where (a) and (b) are any points in the open interval ( (a, b) ).

Here's a more formal proof of the mean value theorem:

Let (f) be a function defined on the open interval ( (a, b)). Suppose that (f) is continuous on ( (a, b)). Then, by the definition of continuous function, the limit of the difference quotient as (h) approaches 0 is equal to the derivative of (f) at (c), which is defined as (f'(c)).

Therefore, it follows that

$\lim_{h \to 0} \frac{f(b) - f(a)}{h} = f'(c)$

which proves that (f'(c) = \frac{f(b) - f(a)}{b - a}).

This theorem has several applications in calculus, including finding critical points of functions, determining whether a function is increasing or decreasing, and finding the average rate of change of a function