How do you find the number c that satisfies the conclusion of the Mean Value Theorem for the function #f(x)=x-2sinx# on the interval [0, pi]?

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Answer 1 · 2017-12-04T16:04:28

Solve the equation #f'(x) = (f(pi)-f(0))/(pi-0)# on the interval #(0,pi)#.

#f'(x) = 1-2cosx#

#(f(pi)-f(0))/(pi-0) = ((pi-2sinpi)-(0+2sin0))/pi = 1#

#1-2cosx = 1# at #x=pi/2 + pik# for integer #k#.

The only solution in #(0,pi)# is #pi/2# so

#c = pi/2# is the only #c# that satisfies the conclusion of the mean value theorem for this function on this interval.