If-x-y-z-gt-0-then-4x-2-x-y-8y-2-y-z-4z-2-z-x-2x-5y-z-

Question Number 154384 by mathdanisur last updated on 17/Sep/21

If x; y; z > 0 then :

\frac{{4 x}^{2}}{x + y} + \frac{{8 y}^{2}}{y + z} + \frac{{4 z}^{2}}{z + x} ⩾ 2 x + 5 y + z

Answered by ghimisi last updated on 18/Sep/21

4 (\frac{x^{2}}{x + y} + \frac{y^{2}}{y + z} + \frac{y^{2}}{y + z} + \frac{z^{2}}{x + z}) ⩾ 4 \cdot \frac{{(x + y + y + z)}^{2}}{x + y + y + z + y + z + x + z} =

= 4 \cdot \frac{{(x + 2 y + z)}^{2}}{2 x + 3 y + 2 z} \overset{∙}{⩾} 2 x + 5 y + z

∙ \Leftrightarrow 4 {(x + 2 y + z)}^{2} ⩾ (2 x + 5 y + z) (2 x + 3 y + 2 z) \Leftrightarrow \dots \Leftrightarrow {(y - z)}^{2} ⩾ 0

Commented by mathdanisur last updated on 18/Sep/21

Very nice Ser, thankyou