
Mean Value Theorem
Using the Mean Value Theorem, prove that ln(103)  ln(100) is less than or equal to 0.03
I tried taking f(x) = 1/x on the interval [100,103]. The average value would then be (1/3)[ln(103)ln(100)].
I don't know how to proceed after this or even if I'm on the right track.

Well, you actually gotta take $\displaystyle f(x)=\ln x$ then exists $\displaystyle c\in(100,103)$ so that $\displaystyle \frac{\ln 103\ln 100}{3}=\frac{1}{c}.$
On the other hand is $\displaystyle 100<c<103$ so $\displaystyle \frac1{103}<\frac1c<\frac1{100}$ and then $\displaystyle \frac{1}{103}<\frac{\ln 103\ln 100}{3}<\frac{1}{100},$ so pick the second inequality and $\displaystyle \ln 103\ln 100<\frac{3}{100}=0,03.$