The mean value theorem tells us (roughly) that if we know the slope of the secant line of a function whose derivative is continuous, then there must be a tangent line nearby with that same slope. This lets us draw conclusions about the behavior of a function based on knowledge of its derivative.
Lecture Video and Notes
Clip 1: Description of the Mean Value Theorem
Clip 2: Consequences of the Mean Value Theorem
Increasing or Decreasing
Generalizing the Mean Value Theorem