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# help

give me the total solution

Note by Superman Son
4 years ago

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Diganta B., please don't forget to mention $$\geq$$ instead of $$>$$ in your question.

For the first one: Assume, by symmetry, $$a \geq b \geq c$$. Then, by Rearrangement inequality on the three sets $$a^{2},b^{2},c^{2}$$ , $$a^{3},b^{3},c^{3}$$ and $$a^{3},b^{3},c^{3}$$, we get

$$a^{2}a^{3}a^{3}+b^{2}b^{3}b^{3}+c^{2}c^{3}c^{3}\geq a^{2}b^{3}c^{3}+a^{3}b^{2}c^{3}+a^{3}b^{3}c^{2}$$

which gives $$a^{8}+b^{8}+c^{8} \geq a^{3}b^{3}c^{3} (\frac {1}{a} + \frac {1}{b} + \frac {1}{c})$$ And finally

$$\frac {a^{8}+b^{8}+c^{8}}{a^{3}b^{3}c^{3}} \geq (\frac {1}{a} + \frac {1}{b} + \frac {1}{c})$$

For the second question:

$$\frac {a^{8}+b^{8}+c^{8}}{3} > (\frac {a+b+c}{3})^{8}$$ is the same as

$$(\frac {a^{8}+b^{8}+c^{8}}{3})^{\frac {1}{8}} > \frac {a+b+c}{3}$$ and this nothing but the generalized mean with $$M_{8}(a,b,c) > M_{1}(a,b,c)$$. I hope you know about the generalized mean inequality.

Equality in both cases holds iff $$a=b=c$$ · 3 years, 12 months ago

thanks · 3 years, 12 months ago

could you type these up? no offense, but i can't read any of the exponents · 3 years, 12 months ago

exponent is 8 · 3 years, 12 months ago

use AM-GM inequality · 4 years ago

please explain totally · 4 years ago

sorry lah... ok I'll try shortly · 4 years ago

AM GM means Arithmetic mean and geometric mean...and there is an proved inequality for that... · 3 years, 12 months ago

i know that · 3 years, 12 months ago

i. By AM-GM,

$\frac28a^8+\frac38b^8+\frac38c^8\geq a^2b^3c^3\\ \frac28b^8+\frac38c^8+\frac38a^8\geq b^2c^3a^3\\ \frac28c^8+\frac38a^8+\frac38b^8\geq c^2a^3b^3$

Adding these three inequalities and dividing by $$a^3b^3c^3$$ gives the desired inequality.

ii. This is a special case of the power-mean inequality on three variables. · 3 years, 12 months ago

I think the second one is Titu's Lemma... · 3 years, 12 months ago

1st one is obvious by weighted AM-GM. Second one is obvious by Jensen's inequality. · 3 years, 12 months ago

Please type this up. I can't read it clearly. · 3 years, 12 months ago

explain clearly · 3 years, 12 months ago

Please don't, it is best for people to figure it out for themselves! By giving away more we learn less. · 3 years, 12 months ago

I thought we weren't supposed to put boring homework on this forum... · 4 years ago

it is not my homework guys · 3 years, 12 months ago

its an example from a book called excursion in mathemay=tics · 3 years, 12 months ago

Ya! It is a famous book in India and a must for the basics. · 3 years, 12 months ago

I am sure you did not see the problems properly. · 3 years, 12 months ago