# Thread: proof problem( I dont know what this belong to)

1. ## [SOLVED]proof problem( I dont know what this belong to)

prove lnx < square root of x if x > 0

2. ## ok

ok you have to redefine them....$\displaystyle \ln(x)=\int_1^{x}\frac{1}{y}dy$ and $\displaystyle \sqrt{x}=\int_0^{x}\frac{1}{2y}dy$ $\displaystyle \forall{x}>0$ from there you can see what to do..but then again $\displaystyle \ln(x)\sim\sqrt{x}$...so you could make a bid that this is untrue

3. Originally Posted by Mathstud28
ok you have to redefine them....$\displaystyle \ln(x)=\int_1^{x}\frac{1}{y}dy$ and $\displaystyle \sqrt{x}=\int_0^{x}\frac{1}{2y}dy$ for every $\displaystyle x>0$ from there you can see what to do..but then again $\displaystyle \ln(x)$~$\displaystyle \sqrt{x}$...so you could make a bid that this is untrue
sorry, i misunderstood
can u see this?
http://img99.imageshack.us/img99/5271/13493377zy4.png

this is how to prove by solution manual
I dont understood how the minimum is 4

can u tell me the reason?
thank you

4. $\displaystyle \frac{1}{2\sqrt{x}} - \frac{1}{x} = 0$
$\displaystyle \frac{1}{2 x^{\frac{1}{2}}} = \frac{1}{x}$
$\displaystyle 2x^{\frac{1}{2}} = x$
$\displaystyle 2 = x^{\frac{1}{2}}$
$\displaystyle x = 4$

Use your first derivative test and you'll see it's a local/absolute minimum.

5. ## Ok

if $\displaystyle f(x)=\sqrt{x}-\ln(x)$ then $\displaystyle f'(x)=\frac{1}{2\sqrt{x}}-\frac{1}{x}$ find the places where it is undefined and zero..this is at x=0(undefined) and x=4(zero) since 0 is not in the domain we can get rid of it..now we need to check where the original function is undefined...it is undefined nowhere on the domain of the question...so now all we have is the point x=4...we test that in the second derivative$\displaystyle f''(x)=\frac{1}{x^2}-\frac{1}{4x^{\frac{3}{2}}}$ and we test the point 4 in there..since it gives$\displaystyle \frac{1}{32}$ which is positive $\displaystyle x=4$ is a minimum point on$\displaystyle (0,\infty)$

6. ## Someone good

can you tell me if what I said was correct that since $\displaystyle \ln(x)\sim\sqrt{x}$ that you could make an argument that $\displaystyle \forall{x}>0,\ln(x)<\sqrt{x}$ is untrue?

Edit:forget it I dont know what I was thinking they are not asymptotically equivalent...forget I asked

7. Originally Posted by Mathstud28
$\displaystyle \sqrt{x}=\int_0^{x}\frac{1}{2y}dy$
Careful when workin' with such parameters, this one is wrong.

8. Hi,

Integrate from 1 to x, uh ?

9. ## What?

Integrate from 1 to x for what? and Krizalid is that wrong because the boundary restriction of that does not mesh well with this problem?

10. ..

$\displaystyle \int_0^x \frac{1}{2y} dy = \frac{1}{2} \ln(x) \big|^x_0$

Which is not defined at 0.

Plus $\displaystyle \ln(\sqrt{x})$ is not equal to $\displaystyle \sqrt{x}$

11. ## Oh

I am sorry that is a typo it should read $\displaystyle \sqrt{x}=\int_0^{x}\frac{1}{2\sqrt{y}}dy$...sorry for the typo

12. I think that you should sometimes leave integrals to solve problems the way it's good to solve

13. ## Haha

I am not quite sure what you mean by that..but ok! haha =D