## Mean Value Theorem

### The Mean Value Theorem is a fundamental theorem in calculus that relates the average rate of change of a function to its instantaneous rate of change at a specific point

The Mean Value Theorem is a fundamental theorem in calculus that relates the average rate of change of a function to its instantaneous rate of change at a specific point. It states that if a function f(x) is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in the interval (a, b) such that the instantaneous rate of change of f(x) at c (given by f'(c)) is equal to the average rate of change of f(x) over the interval [a, b] (given by [f(b) – f(a)] / (b – a)).

In simpler terms, it means that if you have a continuous and differentiable function on a closed interval, then at some point within that interval, the instantaneous slope (derivative) of the function will be equal to the average slope between the two ends of the interval.

This theorem is valuable because it guarantees the existence of a specific point where the instantaneous rate of change matches the average rate of change. It has many applications in mathematics, such as in proving the existence of solutions to equations, analyzing the behavior of functions, and optimizing problems. The Mean Value Theorem is also a precursor to other important theorems in calculus, like the Fundamental Theorem of Calculus.

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