Inequalities....... (1 Viewer)

[zeek]

Can anyone offer any advice? is there any kind of pattern to them or should i jst do it through trial and error?

 

[Riviet]

The AM/GM inequality is useful: (a₁+a₂+...+a_n)/n > (a₁a₂...a_n)^1/n

 

[bboyelement]

a^2 + b^2 + c^2 = (a+b+c)^2 - 2(ab+bc+ca)

and the AM/GM

theres not many tips for inequalities you just have to remember how to get the usual ones and the techniques

say
a^2 + b^2 >= 2ab

let a = a/b and b = b/a

(a/b)^2 + (b/a)^2 >= 2

everyone should know that but ... yeh

and some ppl find it easier to go backwards
actually the best advice is just do heaps of question and become familiar with it

 

[A l]

I'm not sure what you're after but when given an inequality to prove, manipulate the given expression until you end up with a something that is true and you can provide reasons for it.
For example:
Prove a² + b² + c² ≥ 2(ab + ac + bc)
Manipulate or play around with the expression and you may get:
a² + b² + c² - 2(ab + ac + bc) ≥ 0
(a + b + c)² ≥ 0
You know this is statement is true for all real a, b and c, so this is your starting point.
(a + b + c)² ≥ 0
a² + b² + c² - 2(ab + ac + bc) ≥ 0
.: a² + b² + c² ≥ 2(ab + ac + bc)

 

[Riviet]

Yeah, going backwards can sometimes be the better option, when you come back to something that's clearly true, cross it out and copy it out forwards!

 

[Affinity]

I think it's more about how good you are with algebraic manipulation