Question

I am so very lost, I believe it dose not apply but I'm unsure. determine whether the mean value theorem to be applied to the function f(x)=2sin(x)+sin2x on the closed interval [7pi,8pi] if the mean value theorem can be applied find all numbers "c" in the open interval (7pi,8pi) such that f'(c)=(f(8pi)-f(7pi))/(8pi-7pi)

Answer 1

To be able to apply the Mean Value Theorem,
the function must be
`continuous` on [7pi,8pi]
`differentiable` on (7pi,8pi)

So what makes you think it can't be applied? :o
The function is certainly continuous, yes? :)
It's just the sum of sines.

Answer 2

The theorem basically says this:

If we have continuity and smoothness of the function,
then there is at least one `tangent line` which has the same slope as the `secant line` which connects the end points of our interval

Answer 3

Do you understand how to find the slope of that secant line? :)

Answer 4

Ok

Answer 5

It should be parallel to the tangent line

Answer 6

Draw a line through point A(a(f(a)) and B(b,f(b))

Answer 7

Good yes! :)
Same slope = parallel lines.
This is an example of what that might look like.
In this example we have multiple tangent lines fulfilling this.