Let f(x) = x + sin2x on [0, 2(pi)] find two numbers c that satisfy the conclusion of hte Mean Value Theorem. (note there are four such numbers.)
$\displaystyle a=0, \;\ b=2\pi$
$\displaystyle f(x)=x+sin(2x)$
$\displaystyle f(a)=0, \;\ f(b)=2\pi$
$\displaystyle f'(x)=2cos(2x)+1$
$\displaystyle f'(c)=2cos(2c)+1$
Also, $\displaystyle f'(c)=\frac{f(b)-f(a)}{b-a}=\frac{2\pi-0}{2\pi-0}=1$
So, we have $\displaystyle 2cos(2c)+1=1$
Solving for c we find $\displaystyle c=\frac{(2c-1){\pi}}{4}$
When c= 1 through 4, we get:
$\displaystyle c=\frac{\pi}{4}, \;\ \frac{3\pi}{4}, \;\ \frac{5\pi}{4}, \;\ \frac{7\pi}{4}$
are the four points which satisfy the MVT over the given interval.