Differential mean value theorem?

Differential mean value theorem?

If the function f (x) is continuous in the closed interval [a, b], differentiable in the open interval (a, b), and the function values at the end of the interval are equal, that is, f (a) = f (b), then there is at least one point ξ (a) in (a, b)

What is differential mean value theorem?

For continuous function f (x), if f (a) = f (b) = 0, then there must be x belonging to (a, b) such that f '(x) = 0;
Or if f (b) ≠ f (a), X must belong to (a, b), such that f (b) - f (a) / B-A = f '(x)
Conditions may not be very strict, you can refer to "higher mathematics" Tongji edition

Why is the differential mean value theorem called mean value theorem? What is the meaning of the word "mean value"? What are the aspects reflected in the theorem? I have been confused about the origin of the name of mean value theorem before I learned it. Maybe I haven't learned it thoroughly and grasped the essence of the theorem

Because the mean value theorem says: in (a, b), there is at least one point ξ such that
This value is a ξ in the interval (a, b), so it is called median