Mean value theorem for integrals.

[bobbarkernar]

find c such that (f)average=f(c)

f(x)=7sin(x)-sin(2x), [0,pi]

ok so i found (f)average= 14/pi

then i tried to compute:
7sin(c)-sin(2c)= (14/pi)

but the answers i got were wrong.
please if someone could reply with some helpful information on how to solve this. thank you

 

[courtrigrad]

this is how i would start it:

$$7\sin c - \sin 2c = \frac{14}{\pi}$$

$$7\sin c - 2\sin c \cos c = \frac{14}{\pi}$$.

Let $$a = \sin c$$ and $$b = \cos c$$.

Then $$7a-2ab = \frac{14}{\pi}$$. We also know that $$a^{2} + b^{2} = 1$$. Can you go from there?