The Mean Value Theorem states that for any continuous function defined on a closed interval, the function takes on every value in the interval at least once...
The Mean Value Theorem states that for any continuous function defined on a closed interval, the function takes on every value in the interval at least once.
In other words, there exists a point c in the interval [a, b] such that f(c) is equal to the average of the function values at the endpoints a and b:
This means that the function takes on the same value at some point in the interval, which can be proved using the definition of the average value.
Examples:
Constant Function: If a function is constant, then it takes on the same value for all values in the interval. The mean value theorem then states that the function takes on that same value at some point in the interval.
Linear Function: A linear function always takes on the same slope (a constant) in any interval. Therefore, the mean value theorem states that it takes on the same value at some point in the interval.
Quadratic Function: A quadratic function takes on a minimum and a maximum value in any interval. The mean value theorem states that it takes on both of these values at some point in the interval