How do you find the number c that satisfies the conclusion of the Mean Value Theorem for the function #f(x)=x/(x+9)# on the interval [1,4]?

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Answer 1 · 2015-07-24T17:45:30

#c=sqrt{130}-9 \approx 2.40175#

Since #f(x)=x/(x+9)#, we have #(f(4)-f(1))/(4-1)=(4/13-1/10)/3=(40-13)/390=27/390=9/130#

Also, #f'(x)=((x+9)*1-x*1)/(x+9)^2=9/(x+9)^2#

Setting #f'(c)=9/130# implies that #(c+9)^2=130# so that #c+9=\pm sqrt{130}# and #c=-9\pm sqrt{130}#.

Of these, only #c=-9+sqrt{130}\approx 2.40175# is in the interval #[1,4]#, so that's the (unique) answer.