Mean value theorem

Theorem in mathematics

In calculus and real analysis, the mean value theorem (or Lagrange's mean value theorem) is a theorem about differentiable functions, roughly stating that the average rate of change of such a function over an interval is equal to the instantaneous rate of change at some point in the interval. For example, if a car smoothly travels a certain distance over a given finite time interval, then at some moment during the trip, its instantaneous speed equals its average speed for the whole trip. The theorem states precisely that if a real-valued function is continuous on a closed interval [ a , b ] {\displaystyle [a,b]} , with a < b {\displaystyle a<b} , and differentiable on the interior ( a , b) {\displaystyle (a,b)} , then there is at least one point in ( a , b) {\displaystyle (a,b)} where the derivative equals the function's average rate of change over the whole interval.

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