How to learn inequalities? (1 Viewer)

[QZP]

Anyone got general tips; im just learning inequalities? It's so hard -_- and I don't know in which direction to think

Edit: Atm I'm just doing mass questions to "memorise" the methods for doing them

 

[QZP]

Also got a Q, am I allowed to just quote the AM >= GM inequality?

 

[emilios]

Dw i'm in the same boat but these are some of the things I've picked up:
- Doing heaps of questions definitely does help. Not really actively memorizing things, but just learning how to recognize patterns.
- Most things start from any variation on (a-b)^2 >= 0 (i.e. change up the values of a and b for different q's)
- AM-GM is everything. You use it in like 80% of q's
- Always look at what they've asked you to just prove. You'll almost always use the result of pt i) in pt ii) or iii) or the result in ii) in iii)... you get the point
- While your proof has to be 'fowards' i.e. usually has to start from scratch, you can always work the question out backwards, get to an obviously true result, then reverse the process and cross out your initial working haha

 

[emilios]

Also got a Q, am I allowed to just quote the AM >= GM inequality?

Probs not. It takes like 2-3 lines to prove, just do it this way;

(sqrt(a) - sqrt(b))^2 >=0
a+b -2sqrt(ab) >= 0
a+b/2 >= sqrt(ab)

 

[emilios]

Oh and also making smart substitutions. Say you've just proved:
a^3 + b^3 + c^3 >= 3abc

Letting x= a^3 , y=b^3 , z= c^3 will lead you to the AM-GM result for n=3

 

[QZP]

Oh and also making smart substitutions. Say you've just proved:
a^3 + b^3 + c^3 >= 3abc

Letting x= a^3 , y=b^3 , z= c^3 will lead you to the AM-GM result for n=3

Yeah I picked up on that one hahaha Thanks !! Raelly appreciate it

 

[iStudent]

Dw i'm in the same boat but these are some of the things I've picked up:
- Doing heaps of questions definitely does help. Not really actively memorizing things, but just learning how to recognize patterns.
- Most things start from any variation on (a-b)^2 >= 0 (i.e. change up the values of a and b for different q's)
- AM-GM is everything. You use it in like 80% of q's
- Always look at what they've asked you to just prove. You'll almost always use the result of pt i) in pt ii) or iii) or the result in ii) in iii)... you get the point
- While your proof has to be 'fowards' i.e. usually has to start from scratch, you can always work the question out backwards, get to an obviously true result, then reverse the process and cross out your initial working haha

And learning the neat little tricks as well! (algebra manipulation, substitution etc)

 

[Kurosaki]

Proof by contradiction is also a powerful tool.

 

[emilios]

Haha I feel so dirty when I do a proof by contradiction. It's like, see those 15 lines of working out? Yeah they're all wrong m8

 

[glittergal96]

I find that most (like 95% of) inequalities we do in the HSC come from squares being non-negative. It's just sometimes not obvious what things to square.

A sometimes useful trick I use is trying to figure out (or guess lol) when equality holds in an inequality. (Because if the inequality is from squares being non-negative, equality occurs when the squares are zero...that is when the thing you are squaring is zero.)

This can get unmanageable if the expressions are complicated though.

 

[Sy123]

Have a look at the Advanced level marathon, a lot of reasonably hard inequality questions are there, and although they may use some slightly more advanced techniques (i.e. General AM-GM, Cauchy and Jensen's inequality), the skill set required translates well into the HSC environment.

A lot of inequalities comes down to 'seeing' what to do, there is no really set procedure to it, that is why BOS uses inequalities (and Combinatorics) as a sort of buffer between medium band 65 and high band 6 students.

Here are some 'techniques'

- Keep 'solving' the inequality by doing things to both sides until you get a result easier to prove
- Try a substitution to lower the amount of variables, potentially allowing you to use calculus if you are able to get it down to 1 variable.

Although substitution is not something BOS will ask out of the blue, it is a good techinque to keep in mind, common ones are:



As you can see, although we still have 3 variables, we get the added product condition which can prove useful

- Keep note of how 'strong' the inequality is

What I mean by this is:

Lets say the question asked you to prove that

A > B

You use AM-GM and you get A > C

And you try to simplify C > B, perhaps to get an answer, and you end up proving a contradiction

Which means B > C

Which means that A > B is an inequality that is 'stronger' than AM-GM, meaning you will probably need to do something more complex

Though writing all these tips down won't really help, you need to do some advanced inequalities to gain the intuition necessary