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

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Answer 1 · 2016-11-06T15:42:45

Solve the equation: #f'(x) = (f(pi/2)-f(-pi/2))/(pi/2-(-pi/2))#. Remember that #c# must be in #(-pi/2,pi/2)#

#f'(x) = 1+sinx#

#(f(pi/2)-f(-pi/2))/(pi/2-(-pi/2)) = 1#

So we want solutions to

#1+sinx = 1# that are in the interval #(-pi/2,pi/2)#

So we need #sinx = 0# which happens at integer multiples of #pi#.

The only integer multiple of #pi# in the interval we're interested in is #0#.

#c=0#