Prove-that-a-4-b-4-c-4-abc-a-b-c-

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

Prove that a^{4} + b^{4} + c^{4} ⩾ a b c (a + b + c)

Answered by mrW1 last updated on 19/Jul/17

a^{4} + b^{4} ⩾ {2 a}^{2} b^{2}

b^{4} + c^{4} ⩾ {2 b}^{2} c^{2}

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

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

a^{4} + b^{4} + c^{4} ⩾ \frac{1}{2} abc (2 b + 2 a + 2 c)

a^{4} + b^{4} + c^{4} ⩾ abc (a + b + c)

Commented by Tinkutara last updated on 19/Jul/17

Thanks Sir!