mean value theorem inequality homework

Question

So I think the interval for this question is (n,n+1) and the function is y=lnx, based what the inequality looks like, but I don't know how to approach this question the "proper" way. Also I am not sure what the (n+1)/n does. so if someone can explain it I would be very happy.

Thanks

$\ln{\left(1+\frac{1}{n}\right)}<\frac{1}{n}$. What does this remind you of when you simplify? — Element118, Dec 11 '15 at 23:10
@Element118 , I don't really know... sorry. But thanks for the input. — Spongebob Squareroot-pants, Dec 11 '15 at 23:14
Can you show that $\left(1+\frac{1}{n}\right)^n$ is increasing? — Element118, Dec 11 '15 at 23:22
@Element118 it would be increasing... but how would it relate though? — Spongebob Squareroot-pants, Dec 11 '15 at 23:52
Do you know that the limit is $e$? What can you conclude by taking the $\log$ of this expression? — Element118, Dec 11 '15 at 23:54
https://math.stackexchange.com/q/652581/321264, https://math.stackexchange.com/q/1887583/321264 — StubbornAtom, Oct 22 '21 at 13:15
Does this answer your question? Intuition behind logarithm inequality: $1 - \frac1x \leq \log x \leq x-1$ — StubbornAtom, Oct 22 '21 at 13:15

score 1 · Accepted Answer · answered Dec 11 '15 at 23:40

1

Hint: $\ln' x=1/x$ and its value is between $1/(n+1)$ and $1/n$.

answered Dec 11 '15 at 23:40

Eugene Zhang

16,805

Yes. so the slope of the secant line connecting the 2 points is between the tangent at the 2 ends. but where does n+1/n come in? – Spongebob Squareroot-pants Dec 11 '15 at 23:49
1

$ln(n+1)-ln(n)=ln(\frac{n+1}{n})$ – mrprottolo Dec 12 '15 at 00:33

mean value theorem inequality homework

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