Question

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

Answer 1

The Mean Value Theorem says that
if a function `f` is
continuous on the closed interval `[a, b]` and
differentiable on the open interval `(a, b)`,

then there is a number `c` in `(a, b)` with

`color(white)"sssssssssssssss"` `f'(c)= (f(b)-f(a))/(b-a)`.

(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:
`f(x)=2sin x + sin 2x` the interval is `[4 pi , 5 pi]` and `(4 pi , 5 pi)` .

It is true that:
`f` is continuous at every real number, so it is continuous on `[4 pi , 5 pi]`
(`sin` is continuous, and `2x` is continuous so `sin 2x` is contiunuous. And the sum of continuous functions is continuous.)

It is also true that
`f` is differentiable at every real number, so it is differentiable on `(4 pi , 5 pi)`.
(Differentiable means the derivative exists and `f'(x)=2cosx+2cos2x` exists (is defined) for all values of `x`.)

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

There is a number `c` in `(4 pi, 5 pi)` with:

`color(white)"sssssssssssssss"` `f'(c)= (f(5 pi)-f( 4 pi))/(5 pi - 4 pi )`.

Doing some arithmetic we can rewrite this conclusion as;

We can conclude that
There is a number `c` in `(4 pi, 5 pi)` with:

`color(white)"sssssssssssssss"` `2cosc+2cos2c`= 0#.

That conclude the use of the Mean Value Theorem for `f(x)=2sin x + sin 2x` on the interval is `[4 pi , 5 pi]`

(As an additional exercise in solving equations, your teacher or textbook may also ask you to find the value or values of `c` that the theorem assures us are there.)