# How do you use the mean value theorem for #2sinx +sin2x# on closed interval of #[4pi, 5pi]#?

##### 1 Answer

The Mean Value Theorem says that

**if** a function

continuous on the closed interval

differentiable on the open interval

**then** there is a number

(Note: The amount of detail you'll be asked to provide depends, to some extent, on your teacher's goals for the class.)

**In the question:**

It is true that:

(

It is also true that

(Differentiable means the derivative exists and

Theerfore, the Mean Value Theorem allows us to conclude that:

There is a number

Doing some arithmetic we can rewrite this conclusion as;

We can conclude that

There is a number

That conclude the use of the Mean Value Theorem for

(As an **additional** exercise in solving equations, your teacher or textbook may also ask you to find the value or values of