Question

Just another super easy problem,

Let \(x, y, z \in \mathbb{R}^+\) such that \(x +y +z =\sqrt{3} \) . Find the maximum value of \[ \frac x{\sqrt{x^2+1}} +\frac y{\sqrt{y^2+1}} +\frac z{\sqrt{z^2+1}} \]

Answer 1

It wouldn't just happen to be 1.5 would it?

Answer 2

Yes, what did you used ? ;)

Answer 3

Lucky guess ;)

Answer 4

lol, The options given were tricky so guessing won't work :( The actual solution is very cute and enlightening :)

Answer 5

yeah I see it as there being two options, the boundaries of \(x+y+z=\sqrt3\) or the average where \(x=y=z=\frac{\sqrt3}3\)
what kind of proof you want that this is the maximum I'm not sure; you could use calculus I would imagine

Answer 6

cross multiply

Answer 7

x(y+z) + y(x+z) + z(x+y)