AM-GM x^2+y^2+z^2 >= xy+xz+yz

Guztea Reynaldi

Kak kalo  x^2+y^2+z^ 2 >= xy+yz+xz
dijadiin Am-Gm gmn?

 

Della Faradila
Maksud kmu pmbuktiannya ya?.. Hm mungkin gini kali ya,
GM<=AM
2(Vab) <= a+b
xy+yz+xz <= x^2+y^2+z^2

kalikan kedua ruas dgn 2
2xy+2yz+2xz <= 2x^2+2y^2+2z^2

misal x=Va, y=Vb
a=x^2, b=y^2
2(Vab) <= a+b

2xy <= x^2+y^2
2yz <= y^2+z^2
2xz <= x^2+z^2
————— +
2xy+2yz+2xz =xy +yz+xz

x^2+y^2+z^2=1/2 x^2 +1/2 x^2 + 1/2 y^2+ 1/2 y^2+1/2 z^2 +1/2z^2
x^2+y^2+z^2=1/2 x^2+ 1/2 y^2 + 1/2 y ^2 +1/2 z^2+1/2 x^2+ 1/2 z^2
x^2+y^2+z^2=1/2 (x^2 +y^2)+1/2(y^2+z^2)+1/2(x^2+ z ^2)>=1/2(2xy) +1/2(2 yz)+1/2(2xz)
x^2+y^2+z^2>=xy +yz+xz

 

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