Question

How do i find or approximate the points that are guaranteed to exist by the mean value theorem on f(x)=ln2x; [1,e] ?

Answer 1

\[f'(c)=\frac{f(b)-f(a)}{b-a}\]
\[a=1, b=e, f(1)=\ln(2), f(e)=\ln(2e)=\ln(2)+\ln(e)=\ln(2)+1\]
\[f'(x)=\frac{1}{x}\]

Answer 2

so your job is to solve for c
\[\frac{1}{c}=\frac{\ln(2)+1-\ln(2)}{e-1}\]
\[\frac{1}{c}=\frac{1}{e-1}\]

Answer 3

alright, i'll get on that

Answer 4

one step left! take the reciprocal and you are done

Answer 5

okay, i'll piece it together from here, thanks

Answer 6

i just gotta look it all over

Answer 7

the statement of the mvt is easy to read, the hard part is in going from the general
\[f'(c)=\frac{f(b)-f(a)}{b-a}\] to the specific

Answer 8

so it would just be c=e-1?

Answer 9

or c=(c^2)/(e-1)

Answer 10

i got \[c=e-1\]