How do you determine all values of c that satisfy the conclusion of the mean value theorem on the interval [0, pi] for #f(x)=2sin(x)+sin(4x) #?

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Answer 1 · 2017-04-03T17:15:13

You'll need to solve #2cosx+4cos(4x) = 0# on the interval #(0,pi)#.

The conclusion of MVT says there is a #c# in #(0,pi)# with

#f'(c) = (f(pi)-f(0))/(pi-0)#

Since #f(x) = 2cosx+4cos(4x) # and #(f(pi)-f(0))/(pi-0) = 0#

that means that to find #c# we must solve

#2cosx+4cos(4x) = 0#.