No Doubts about the AM-GM inequality

Shifu: Hello Po, it seems you’ve got a good challenge to solve… how do you plan to go about it?

Po: Master shifu, thanks but I guess there must be some tricks to solve it! I guess I should first get the expression in simplest form and then apply the famous AM/GM. I guess this will work?

Shifu: Why to guess, what you can KNOW! Go ahead do it.

Po: so let 1/a = x then the expression be comes,

1/(1+x) + 1/(1+y) + 1/(1+z) = 1 then prove that xyz ≥8

This is much simple equation to solve, Why did you give me a complex expression to start with? Life is already tough why to make it tougher? So now I’ll apply AM/GM and get the answer What do you say?

Shifu: let’s find out then.

Po: Ok, let’s do it:

[1/(1+x) + 1/(1+y) + 1/(1+z)]/3 ≥ cube root[1/(1+x) *1/(1+y) *1/(1+z)] so [1/(1+x) *1/(1+y) *1/(1+z)] ≥ 1/27

thus [(1+x) *(1+y) *(1+z)] ≥27

But how to get [(x) *(y) *(z)] ≥8 ?

Shifu help me please!

Shifu: Yes, but that’s not complicated. First let’s resolve Why we are doing what we are doing?

Po: What? wait I’ve got another idea… I guess this has some other trick? I feel the AG/GM close cousin HM has a role here!

Shifu: What-if it does? What-if does not? and WHY? For the time being let’s assume it does? So go ahead and invite mr. HM in your calculation.

Po: yeah sure but HM’s are a bit lengthy and confusing? I’ve a doubt why this Q is so complex?

Shifu: Nothing is complex if you know the insight… and remember there are no tricks only insights!

Po: Oh Yeah! As in “There are NO accidents” Right? But Why do I only get these complex Qs to solve.
Shifu: Focus PO, go ahead give it one more try? If you have any doubts ask, I’m here to discuss.

Po: OK!

What if we do not simply the expression but re-write in a,b,c only as {1/ 1+(1/a)} + {1/ 1+(1/b)}+{1/ 1+(1/c)} = 1 and apply he GM/HM inequality?

Shifu: Fair enough give it a sincere try!

PO: Okay, I’m feeling hungry now! Shall I send it to SolveitNOW?
Shifu: Focus, and don’t go for easy way out! Give it another try, you’re doing good!

PO: God please help!

So by applying GM≥HM for (1, a) we get root(a) ≥ 2/1+(1/a) and similarly for b and c thus root(a)+root(b)+root(c) ≥ 2( {1/ 1+(1/a)} + {1/ 1+(1/b)}+{1/ 1+(1/c)}

Thus root(a)+root(b)+root(c) ≥ 2

and now if we apply AM/GM then we get [root(a)+root(b)+root(c)] ≥ 3* 6th root of(abc)…. so guess now it’s clear that (abc)>8

Shifu: Is it? How? let’s say [root(a)+root(b)+root(c)] = S and (a*b*c) = P so this HM/GM — AM/GM calculation says S≥2 and S ≥ 3* 6th root of P It does not establish whatsoever any relation for definiteveness of (P).

Po: Okay, so now I’m vexed with this problem. Shall I SolveitNOW? I’m in no mood to torture myself anymore!
Shifu: Very fine! Go ahead also send them your attempts so that they might help you refine your approach.
Po: Umm…Picture Click done… and here you Go SolveitNOW! Let’s go grab some bite master…

Shifu: Po, I’m happy that you’re not quitting on your weak areas. And this willingness to go extra mile to fix your approach and get better, This is the GrowthMindset you need to forge!

A bit later

Po: Master, Master, I got a solution on SolveitNOW, it’s in just one page for which we filled up 10 pages!
Shifu: Po, those close to a dozen pages of exploration are more important than just a 1 page solution. In fact your 10 page exploration helped more of your brain cells and also helped the SolveitNOW Grandmaster to get to the right Solution. Anyway What’s the way to solve .. did you do it yourself, what was the line of thinking you avoided or got missed?

Po: It’s the same AM/GM but a very smart way to apply it, The grandmaster has simplified the expression to the core then expanded the requied terms and solved it in jiffy!

Shifu: OK then you should do it on your own and reflcet on it, show me the working!
PO: Here you go sir!

let a/1+a = x then a = x/(1-x) and similarly the the expression becomes,

x+y+z = 1 then prove that [x/(1-x)]*[y/(1-y)]*[z/(1-z)] ≥8

Let P = [x/(1-x)]*[y/(1-y)]*[z/(1-z)] = xyz/(1-x-y-z+xy+xz+yz -xyz)

now as x+y+z = 1 we get the

P = xyz/(xy+xz+yz -xyz) = 1/[ 1/x + 1/y+1/z -1]

Now when we apply AM/GM then we get

x+y+z ≥ 3 * cube root(xyz)

1/x + 1/y+1/z ≥ 3* cube root( 1/xyz)

Let R =cube root(xyz) and S = 1/x + 1/y+1/z then

1≥ 3 R and S ≥ 3/R thus R≤1/3 and S≥9

Thus the required expression P = 1/S-1 thus P≤ 1/ 9–1 hence

the answer P ≤ 1/8

Shifu: Correct. You just got it right!

Po: That’s the same thing we had done before but missed certain critical points and that led to wrong paths of solution.

Shifu: Yes, A bit of strategic thinking before starting to solve the problem and applying the How-to-Solveit method of SolveitNOW helps you get on with the the right way to solve. Great working today PO! Keep it up!

—brought to you by my.SolveitNOW.org, adapted from “Study guide to Fundamentals of Physics” by Thomas E. Barret

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