Mean value theorem

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 ${\displaystyle [a,b]}$, with ${\displaystyle a<b}$, and differentiable on the interior ${\displaystyle (a,b)}$, then there is at least one point in ${\displaystyle (a,b)}$ where the derivative equals the function's average rate of change over the whole interval. Geometrically, this means that at some point the tangent to the graph is parallel to the secant line through the interval's endpoints. It is used in proving other general properties of differentiable functions.