Mathematics

The Mean Value Theorem guarantees that somewhere inside an interval, the instantaneous rate of change (derivative) must equal the average rate of change over the whole interval. Here you'll find that special point by setting $f'(x)$ equal to the given average value and solving for $x$. Since $f(x) = \ln x$, you already know the derivative—just equate it to $\frac{2}{e^2-1}$ and isolate $x$.

The Mean Value Theorem guarantees that somewhere inside an interval, the instantaneous rate of change (derivative) must equal the average rate of change over the whole interval. Here you'll find that special point by setting $f'(x)$ equal to the given average value and solving for $x$. Since $f(x) = \ln x$, you already know the derivative—just equate it to $\frac{2}{e^2-1}$ and isolate $x$.