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

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Answer 1 · 2016-04-07T11:41:20

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

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

there is a c in $(1, 2)$ $(1,2)$ 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 $1/ln2$ .

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