Question

could anyone explain Mean Value Theorem

Answer 1

I'm sure you have read the text of the Mean Value Theorem.

It is important to have a picture in mind when you are dealing with it.

If you look at it, you will see a graph of a function f on a certain interval.
Its endpoints are A and B.

Besides the graph of f, there is also a straight line that goes from A to B.
The x-coordinate of A is a, the x-coordinate of B is b.

So far everything is clear, I think.

Now look at the other (tangent) line I drawed. It is parallel to the first. This means it has the same slope.

The Mean Value Theorem says just this:

If f is continuous on [a, b],
and it is differentiable on (a, b),
then there is a point C on the graph, somewhere between A and B where the slope is the same as the slope of the "direct route" from A to B.

In mathematical language: \(\dfrac{f(b)-f(a)}{b-a}=f'(c)\)

I hope this helps a little!