Question

Given 3 number such that   : x + y + z =3

 Find the minimum P = xy + yz + zx

Hihi kb vs Linh nah m. n :3

<3 Girl 2k5 - FA

Answer 1

We have:

\(\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2\ge0\forall x,y,z\)

\(\Leftrightarrow2x^2+2y^2+2z^2-2xy-2yx-2zx\ge0\)

\(\Leftrightarrow x^2+y^2+z^2-xy-yx-zx\ge0\)

\(\Leftrightarrow\left(x+y+z\right)^2\ge3\left(xy+yz+zx\right)\Leftrightarrow3\ge xy+yz+zx\)

"=" happen only and only x=y=z=1