mathlover1 said:For all positive real numbers [tex]x,y[/tex] prove that:
[tex] \frac{1}{1+\sqrt{x}}+\frac{1}{1+\sqrt{y}} \geq \frac{2\sqrt{2}}{1+\sqrt{2}}[/tex]
ohubrismine said: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.
That's why i am asking to you guys!Tedjn said:Since x+y=1, y=1-x. Now, you are looking for the minimum. How would you do it?
"For all positive real numbers" means that the statement or equation is true for any number greater than 0 that can be expressed as a decimal or fraction. It includes all numbers between 0 and infinity, but does not include negative numbers.
When conducting scientific research, it is important to consider all possible values and scenarios. By including "for all positive real numbers" in our statements or equations, we ensure that our findings and conclusions are applicable to a wide range of values and not just limited to a specific set of numbers.
No, "for all positive real numbers" and "for all real numbers" are not interchangeable. While "for all real numbers" includes all positive numbers, it also includes negative numbers and zero. This can significantly change the outcome or validity of a statement or equation.
"For all positive real numbers" includes all numbers between 0 and infinity, including fractions and decimals. "For all positive integers" only includes whole numbers that are greater than 0. This means that "for all positive integers" is a subset of "for all positive real numbers".
Yes, "for all positive real numbers" can be proven mathematically using various methods such as induction, contradiction, or direct proof. However, it is important to note that the proof may vary depending on the specific statement or equation being examined.