- #1
bill01
Is it possible to prove this:
x^x + y^y < (x+y)^(x+y) for every x,y >=1 ?
x^x + y^y < (x+y)^(x+y) for every x,y >=1 ?
Proving an inequality means demonstrating that one mathematical expression is greater than, less than, or equal to another expression. This is often done using algebraic manipulations and logical reasoning.
The direction of the inequality is often indicated by the given problem or by the context of the situation. For example, if the problem asks for the largest possible value of a certain expression, you would need to prove the inequality in the direction of proving the expression is less than or equal to a certain value.
Some common techniques include algebraic manipulations (such as factoring, expanding, or simplifying expressions), using known properties and theorems (such as the triangle inequality or the AM-GM inequality), and utilizing logical reasoning (such as proving by contradiction or using counterexamples).
Yes, it is important to show all the steps in an inequality proof to ensure that the logic is clear and correct. Skipping steps or assuming certain steps without showing them can lead to errors in the proof.
Some tips include clearly defining the variables and conditions, trying different approaches if you get stuck, and checking your work for errors. It is also helpful to get familiar with common inequality properties and theorems. Practice and patience are key in mastering the skill of proving inequalities.