How do you determine all values of c that satisfy the conclusion of the mean value theorem on the interval [1, 18] for #f(x)=x/(x+9)#?

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Answer 1 · 2016-10-17T20:04:19

Solve #f'(x)=(f(18)-f(1))/(18-1)#. Discard solutions outside #(1,18)#.

#f'(x) = 9/(x+9)^2#

#(f(18)-f(1))/(18-1) = 1/30#

#9/(x+9)^2 = 1/30# has solutions

#x=-9+-3sqrt30#

The only solution in #(1,18)# is #-9 + 3sqrt30# so that is the only #c# that satisfies the conclusion of the MVT for this function on this interval.