# For all positive real numbers

1. Jun 22, 2010

### mathlover1

For all positive real numbers $$x,y$$ prove that:

$$\frac{1}{1+\sqrt{x}}+\frac{1}{1+\sqrt{y}} \geq \frac{2\sqrt{2}}{1+\sqrt{2}}$$

2. Jun 22, 2010

### ohubrismine

Re: inequality

No proof is possible.
Let x=y=1, then left hand side is equal to 1 which is strictly less than the right hand side.

3. Jun 22, 2010

### mathlover1

Re: inequality

I forgot to say that x,y are numbers such that x+y=1

4. Jun 22, 2010

### Tedjn

Re: inequality

Since x+y=1, y=1-x. Now, you are looking for the minimum. How would you do it?

5. Jun 22, 2010

### mathlover1

Re: inequality

That's why i am asking to you guys!
This inequality is correct .. just need to be proven!

6. Jun 22, 2010

### Staff: Mentor

Re: inequality

Find the minimum value of
$$f(x) = \frac{1}{1 + \sqrt{x} } + \frac{1}{1 + \sqrt{1 - x}}$$

This involves finding the critical points.