## clara1223 one year ago Determine if the Mean Value Theorem applies to the function f(x)=2sin(x)+sin(2x) on the interval [0,pi]. If so, find all numbers c on the interval that satisfy the theorem.

1. zepdrix

Which one is the Mean Value Theorem again? Oh right right right.. the one with the uhhhh... You have some interval... With the secant line connecting the end points... Mean Value Theorem tells us that there is a tangent line somewhere inside of that interval. So ummm

2. clara1223

yeah so you have to find (f(pi)-f(0))/(pi-0)

3. clara1223

and then set that equal to the derivative of the function

4. clara1223

i just have trouble with trig on this kind of problem

5. ganeshie8

how do you know MVT can be applied here? have you shown that the given function meets the requirements necessary for applying MVT ? |dw:1444089635169:dw|

6. clara1223

I graphed it and it is continuous and differentiable on the interval

7. ganeshie8

Fair enough! now you're ready to apply MVT

8. ganeshie8

As a start, find the slope of secant line of f(x)=2sin(x)+sin(2x) between [0,pi]

9. ganeshie8

f(x)=2sin(x)+sin(2x) f(pi) = 0 f(0) = 0 so, clearly (f(pi)-f(0))/(pi-0) = 0

10. ganeshie8

so, you want to solve $2\cos(c)+\cos(2c)=0$ over the interval $$(0,\pi)$$

11. ganeshie8

recall the identity : $$\cos(2x)=2\cos^2x-1$$

12. clara1223

and how do i use that identity?

13. ganeshie8

$$2\cos(c)+2\cos(2c)=0$$ $$\cos(c)+\cos(2c)=0$$ Applying double angle identity : $$\cos(c)+2\cos^2(c)-1=0$$ $$2\cos^2(c)+\cos(c)-1=0$$ Try factoring it