I’m an 8th grader. After browsing aops.com, a math contest website, I’ve seen a lot of problems solved by Cauchy Schwarz. I’m only in geometry (have not started learning trigonometry yet). So can anyone explain Cauchy Schwarz in layman’s terms, as if you are explaining it to someone who has just started geo in 8th grade?
In geometry terms that you can understand, the Cauchy-Schwarz inequality says that:
Among all the parallelograms with sides a and b, the rectangle is the one with the largest area.
Usually you can use this inequality when you are looking for an upper (or lower) bound of an expression.
I wanted to give you an example, but my geometry studies are too far to remember an easy demonstration. Maybe you can ask somebody to give you a compass-and-straightedge demonstration of the equivalence between Cauchy-Schwarz and triangle inequalities.