Question

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.

Answer 1

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

Answer 2

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

Answer 3

and then set that equal to the derivative of the function

Answer 4

i just have trouble with trig on this kind of problem

Answer 5

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

Answer 6

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

Answer 7

Fair enough! now you're ready to apply MVT

Answer 8

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

Answer 9

f(x)=2sin(x)+sin(2x)
f(pi) = 0
f(0) = 0

so, clearly (f(pi)-f(0))/(pi-0) = 0

Answer 10

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

Answer 11

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

Answer 12

and how do i use that identity?

Answer 13

\(2\cos(c)+2\cos(2c)=0\)
\(\cos(c)+\cos(2c)=0\)

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

Try factoring it