let-x-y-z-gt-0-such-that-x-4-y-4-z-4-x-2-y-2-z-2-find-the-minimum-of-the-expression-P-x-2-y-y-2-z-z-2-x-

Question Number 162788 by HongKing last updated on 01/Jan/22

let x; y; z > 0

such that x^{4} + y^{4} + z^{4} = x^{2} + y^{2} + z^{2}

find the minimum of the expression :

P = \frac{x^{2}}{y} + \frac{y^{2}}{z} + \frac{z^{2}}{x}

Answered by alephzero last updated on 01/Jan/22

x^{4} + y^{4} + z^{4} = x^{2} + y^{2} + z^{2}

(x_{1}; y_{1}; z_{1}) = (- 1; - 1; - 1)

(x_{2}; y_{2}; z_{2}) = (0; 0; 0)

(x_{3}; y_{3}; z_{3}) = (1; 1; 1)

But x; y; z > 0

\Rightarrow (x; y; z) = (1; 1; 1)

P = \frac{x^{2}}{y} + \frac{y^{2}}{z} + \frac{z^{2}}{x} = \frac{1^{2}}{1} + \frac{1^{2}}{1} + \frac{1^{2}}{1} = 1 + 1 +

+ 1 = 3

P = 3

Commented by HongKing last updated on 01/Jan/22

My dear Sir thank you, but

need to prove P ⩾ 3 for all x; y; z

such that x^{4} + y^{4} + z^{4} = x^{2} + y^{2} + z^{2}