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

$$\mathrm{Prove}\mathrm{that}{a}^4+b^4+c^4⩾abc(a+b+c)$$

Answered by mrW1 last updated on 19/Jul/17

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

Commented by Tinkutara last updated on 19/Jul/17

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

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