How can I prove a^3+b^3+c^3>=3abc using w=(abc)^1/3/?

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Discussion Overview

The discussion revolves around proving the inequality \( a^3 + b^3 + c^3 \geq 3abc \) using the expression \( w = (abc)^{1/3} \). Participants explore various approaches and considerations related to this inequality, including its validity for different ranges of values for \( a, b, c \), and connections to other mathematical identities.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant mentions having proven \( a^4 + b^4 + c^4 + d^4 = 4abcd \) and seeks guidance on proving the inequality using \( w = (abc)^{1/3} \).
  • Another participant questions the validity of the earlier identity by providing specific values for \( a, b, c, d \) that do not satisfy it, expressing confusion.
  • A participant suggests that the inequality may need to be proven for positive numbers only, noting a counterexample with negative values.
  • There is a proposal to consider different cases based on whether the numbers are less than or greater than 1.
  • One participant clarifies their intent to prove the inequality using \( a^4 + b^4 + c^4 + d^4 = 4abcd \) and emphasizes the importance of focusing on positive numbers.
  • Another participant hints at a proof they plan to post, indicating progress in their understanding.
  • A mathematical expression involving \( 4xyzw \leq x^4 + y^4 + z^4 + w^4 \) is introduced, suggesting a potential pathway to the proof.

Areas of Agreement / Disagreement

Participants express uncertainty about the conditions under which the inequality holds, particularly regarding the values of \( a, b, c \). There is no consensus on a definitive approach or solution, and multiple viewpoints are presented.

Contextual Notes

Some participants note the need to clarify assumptions about the positivity of the variables involved and the implications of using specific mathematical identities in the proof process.

StonedPanda
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So I've been thinking about this for a while for an analysis class. I proved that a^4 + b^4 + c^4 + d^4 = 4abcd. Now I'm supposed to prove the inequality above using w=(abc)^1/3/. I'm not asking anyone to do my homework for me, but maybe someone could point me in the right direction?

Nice to be back here.
 
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i realize i might have posted this in the wrong place. I'm sorry. please move it, although i do think that it is a nice topic for this section.
 
Although this is slightly off-topic I don't how:

a^4 + b^4 + c^4 + d^4 === 4abcd

Let:
a = 1
b = 2
c = 4
d = 7

1^4 + 2^4 + 4^4 + 7^4 = 2674

However:
4*1*2*4*7 = 224

I'm confused :S
 
StonedPanda said:
So I've been thinking about this for a while for an analysis class. I proved that a^4 + b^4 + c^4 + d^4 = 4abcd. Now I'm supposed to prove the inequality above using w=(abc)^1/3/. I'm not asking anyone to do my homework for me, but maybe someone could point me in the right direction?

Nice to be back here.
This inequality doesn't hold for a=-1,b=-1,c=1 so I assume that it must be proven for positive numbers only? About the other response, I assume that you meant to type ">= 4abcd"; is this correct? My suggestion is to consider separately the situation where the numbers are less than 1 from what happens as the numbers a,b,c,d become greater than 1. Also what if you rewrote the equation so that one side would be a more definite quantity and still managed to use w somehow. Just some thoughts.
 
I meant prove a^3+b^3+c^3>=3abc using w=(xyz)^(1/3)

and yes i did mean earlier >=
 
argh i meant using a^4 + b^4 + c^4 + d^4 = 4abcd and d=(abc)^(1/3) prove a^3 + b^3 + c^3 + d^3 = 3abc for positive numbers a,b,c
 
Last edited:
i got it, will post proof tomorrow.
 
4xyzw<=x^4+y^4+z^4+w^4
(((( w=(xyz)^(1/3) ))))
4(xyz)^(4/3)=x^4+y^4+z^4+(xyz)^(4/3)
3(xyz)^(4/3)=x^4+y^4+z^4
((((a,b,c=x^4,y^4,z^4 respetively))))
3(abc)^(1/3)<= a + b + c
and now use a,b,c = j^3,k^3,l^3
 

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