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

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Answer 1 · 2016-04-07T11:41:20

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

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#.