anonymous 4 years ago Just another super easy problem, Let $$x, y, z \in \mathbb{R}^+$$ such that $$x +y +z =\sqrt{3}$$ . Prove that the maximum value of $$\large \frac x{\sqrt{x^2+1}} +\frac y{\sqrt{y^2+1}} +\frac z{\sqrt{z^2+1}}$$ is $$\frac 32$$

1. anonymous

And yes we don't need to use Lagrange multiplier. ( http://en.wikipedia.org/wiki/Lagrange_multiplier ) otherwise it will be boring.

2. asnaseer

we could write:$\frac{x}{\sqrt{x^2+1}}=\sqrt{\frac{x^2}{x^2+1}}=\sqrt{\frac{x^2+1-1}{x^2+1}}=\sqrt{1-\frac{1}{x^2+1}}$so then we want to maximize:$\sqrt{1-\frac{1}{x^2+1}}+\sqrt{1-\frac{1}{y^2+1}}+\sqrt{1-\frac{1}{z^2+1}}$

3. asnaseer

which means we need to maximize each of the denominators

4. asnaseer

which gives me the /gut/ feeling that x=y=z is the solution since we have a constraint on the sum of x, y and z

5. asnaseer

but I cannot prove my /gut/ is correct :)

6. anonymous

The reason why I liked this problem is the fact that it uses an inequality which is very useful in general.

7. asnaseer

is it worth pursuing: let x = a$$\sqrt{3}$$ let y = b$$\sqrt{3}$$ let z = c$$\sqrt{3}$$ so we have: a + b + c = 1

8. blockcolder

AMGM? Cauchy-Schwarz? Chebyshev? (These are the only ones I know.)

9. anonymous

No.

10. asnaseer

I guess thats a no to both blockcolder and myself :(

11. anonymous

I like the way asnaseer approaches any problem, always starting from scratch :)

12. anonymous

I was talking about Jensen's Inequality ( http://en.wikipedia.org/wiki/Jensen's_inequality )

13. anonymous

$x \sqrt{y ^{2}+1}\sqrt{z ^{2}+1} = x \sqrt{(y ^{2}+1)(z ^{2}+1})$

14. anonymous

$x \sqrt{y ^{2}+y ^{2}z ^{2}+z ^{2}+y ^{2}z ^{2}}$

15. anonymous

Your a liar, This is hard

16. blockcolder

Oh, jensen's. Therefore, asnaseer's approach is worth looking at.

17. anonymous

x(y+z) do the same thing for all of them

18. anonymous

If you know jensen's this is just there lines.

19. asnaseer

never had the pleasure of meeting him :)

20. anonymous

yeah you guys can just leave me do this alone

21. anonymous

You were here at 1906 ? :P

22. asnaseer

of course - weren't you?

23. asnaseer

19:06 == 6 minutes past 7pm

24. anonymous

xy + zy + yx + zy + zx + zy 2zx + 2xy + 2zy 2y(x+z) + 2zx x+z = sqrt 3 - y

25. anonymous

Haha :D

26. anonymous

$2({\sqrt{3}-y}) + 2x(\sqrt{3} - x -y)$

27. anonymous

$2\sqrt{3} - 2y + 2{\sqrt{3}}x - 2x ^{2} - 2xy$

28. anonymous

differentiate to find max value

29. anonymous

I'm exhausted

30. blockcolder

Jensen's goes like this, right? $\lambda_1+\lambda_2+\cdots+\lambda_n=1\\ f''(x)>0\\ \therefore f(\lambda_1x_1+\lambda_2x_2+\cdots+\lambda_nx_n)\leq\lambda_1f(x_1)+\lambda_2f(x_2)+\cdots+\lambda_nf(x_n)$

31. asnaseer

I /think/ you are correct blockcolder from what I've just read about this method, but I don't understand it well enough to apply it here.

32. anonymous

This is actually concave. f''(x)<0

33. asnaseer

I /think/ the sum pf your lambda's equate equate to: a+ b + c = 1

34. asnaseer

*of

35. anonymous

I was successful in avoiding the Hessian by keeping things only to one variable.

36. asnaseer

could we divide everything by $$\sqrt{3}$$ to get:$x'+y'+z'=1$where:$x'=\frac{x}{\sqrt{3}}$etc

37. asnaseer

so then we need to maximize:$\frac{x'}{\sqrt{x'^2+3}}+...$

38. asnaseer

sorry, that should be - we need to maximize:$\frac{x'\sqrt{3}}{\sqrt{3x'^2+1}}+...$

39. asnaseer

so now the x', y' and z' correspond to your lambda's blockcolder

40. asnaseer

I'm probably way off the mark as its very late here and I need to get some sleep. good like with this guys. I'll check-in sometime tomorrow (or should I say later today) to how how this was solved.