Question

Can someone explain the mean value theorem to me?

Answer 1

the mean value theorem is a more general version of a more specific theorem called Rolle's theorem and maybe that one would be a little easier to understand.
if we have a differentiable function, then it is continuous. Rolle's theorem essentially says that if the function attains the same value at two different spots then at some point between those two points the derivative is 0.

here are a few cases you can think about


note in both cases we have the same y value reoccurring, and in both cases there is a spot where the tangent has a lope of 0.

Answer 2

so that is for when we have two points with the same y value
but what if we have 2 different points completely
example


the mean value theorem just states that ONE such tangent line will exist for a differentiable function between two points