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

[StonedPanda]

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.

 

[StonedPanda]

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.

 

[tomfitzyuk]

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

 

[ramsey2879]

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.

 

[StonedPanda]

I meant prove a^3+b^3+c^3>=3abc using w=(xyz)^(1/3)

and yes i did mean earlier >=

 

[StonedPanda]

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

 

[StonedPanda]

i got it, will post proof tomorrow.

 

[StonedPanda]

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