How do you determine all values of c that satisfy the conclusion of the mean value theorem on the interval [1,4] for #f(x)=x/(x+9)#?

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Answer 1 · 2016-11-20T18:55:25

Solve #f'(x) = (f(4)-f(1))/(4-1)# on the interval #(1,4)#

#9/(x+9)^2 = ((4/13)-(1/10))/(4-1)#

#9/(x+9)^2 = 9/130#

#x = -9 +-sqrt130#

Clearly, #-9-sqrt(130)# is not in the interval,

so the only value of #c# that satisfies the conclusion of MVT is

#c = -9+sqrt130#.