How do you determine all values of c that satisfy the mean value theorem on the interval [1,2] for f(x) = ln(x)?

1 Answer
Apr 7, 2016

I think the intended question is "Find all values of c that satisfy the conclusion of the mean value theorem . . . "

Explanation:

For f(x) = lnx on [1,2], the conclusion of the Mean Value Theorem says:

there is a c in (1,2) such that (or "for which") f'(c) = (f(2)-f(1))/(2-1).

To find the values of c, we need to solve the equation:

1/x = (ln2-ln1)/(2-1)
discarding any solutions outside (1,2).

We get x=1/ln2.

The c we are looking for is 1/ln2.