1 Determine whether the Mean Value Theorem can be applied to

1) Determine whether the Mean Value Theorem can be applied to the function f(x)= 2sinx+sin2x on the closed interval [ 6pi, 7pi]. If the Mean Value Theorem can be applied, find all numbers c in the open interval (6pi, 7pi) such that f \'(c)=( f(7pi)-f(6pi) )/(7pi-6pi). (I know the answer is MVT applies; 19pi/3 ; I just want to know how they got the answer.)

Solution

Since both endpoints are continuous and differentiable,we can find the derivative of f(c) as
f\'(c) = ((2sin(7?) + sin(2 * 7?)) - (2sin(6?) + sin(2 * 6?)))/?
= 0
Then, differentiate f(x) to get:
f\'(x) = 2cos(x) + 2cos(2x)

If f\'(c) = 0, the possible solutions are as below
0 = 2cos(c) + 2cos(2c)
0 = 2cos(c) + 2(2cos