Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 18369 by Tinkutara last updated on 19/Jul/17

Prove that a^4  + b^4  + c^4  ≥ abc(a + b + c)

$$\mathrm{Prove}\:\mathrm{that}\:{a}^{\mathrm{4}} \:+\:{b}^{\mathrm{4}} \:+\:{c}^{\mathrm{4}} \:\geqslant\:{abc}\left({a}\:+\:{b}\:+\:{c}\right) \\ $$

Answered by mrW1 last updated on 19/Jul/17

a^4 +b^4 ≥2a^2 b^2   b^4 +c^4 ≥2b^2 c^2   c^4 +a^4 ≥2c^2 a^2   a^4 +b^4 +c^4 ≥a^2 b^2 +b^2 c^2 +c^2 a^2   a^4 +b^4 +c^4 ≥abc(((ab)/c)+((bc)/a)+((ca)/b))  a^4 +b^4 +c^4 ≥(1/2)abc(((ab)/c)+((bc)/a)+((ab)/c)+((ca)/b)+((bc)/a)+((ca)/b))  a^4 +b^4 +c^4 ≥(1/2)abc(2b+2a+2c)  a^4 +b^4 +c^4 ≥abc(a+b+c)

$$\mathrm{a}^{\mathrm{4}} +\mathrm{b}^{\mathrm{4}} \geqslant\mathrm{2a}^{\mathrm{2}} \mathrm{b}^{\mathrm{2}} \\ $$$$\mathrm{b}^{\mathrm{4}} +\mathrm{c}^{\mathrm{4}} \geqslant\mathrm{2b}^{\mathrm{2}} \mathrm{c}^{\mathrm{2}} \\ $$$$\mathrm{c}^{\mathrm{4}} +\mathrm{a}^{\mathrm{4}} \geqslant\mathrm{2c}^{\mathrm{2}} \mathrm{a}^{\mathrm{2}} \\ $$$$\mathrm{a}^{\mathrm{4}} +\mathrm{b}^{\mathrm{4}} +\mathrm{c}^{\mathrm{4}} \geqslant\mathrm{a}^{\mathrm{2}} \mathrm{b}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} \mathrm{c}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} \mathrm{a}^{\mathrm{2}} \\ $$$$\mathrm{a}^{\mathrm{4}} +\mathrm{b}^{\mathrm{4}} +\mathrm{c}^{\mathrm{4}} \geqslant\mathrm{abc}\left(\frac{\mathrm{ab}}{\mathrm{c}}+\frac{\mathrm{bc}}{\mathrm{a}}+\frac{\mathrm{ca}}{\mathrm{b}}\right) \\ $$$$\mathrm{a}^{\mathrm{4}} +\mathrm{b}^{\mathrm{4}} +\mathrm{c}^{\mathrm{4}} \geqslant\frac{\mathrm{1}}{\mathrm{2}}\mathrm{abc}\left(\frac{\mathrm{ab}}{\mathrm{c}}+\frac{\mathrm{bc}}{\mathrm{a}}+\frac{\mathrm{ab}}{\mathrm{c}}+\frac{\mathrm{ca}}{\mathrm{b}}+\frac{\mathrm{bc}}{\mathrm{a}}+\frac{\mathrm{ca}}{\mathrm{b}}\right) \\ $$$$\mathrm{a}^{\mathrm{4}} +\mathrm{b}^{\mathrm{4}} +\mathrm{c}^{\mathrm{4}} \geqslant\frac{\mathrm{1}}{\mathrm{2}}\mathrm{abc}\left(\mathrm{2b}+\mathrm{2a}+\mathrm{2c}\right) \\ $$$$\mathrm{a}^{\mathrm{4}} +\mathrm{b}^{\mathrm{4}} +\mathrm{c}^{\mathrm{4}} \geqslant\mathrm{abc}\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right) \\ $$

Commented by Tinkutara last updated on 19/Jul/17

Thanks Sir!

$$\mathrm{Thanks}\:\mathrm{Sir}! \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com