Question

Mă puteți ajuta, vă rog? ​

Answer 1

a)

$\frac{x}{y} + \frac{y}{x} \geq 2 \\\frac{x^{2} + y^{2} }{xy} \geq 2\\x^{2} + y^{2} \geq 2xy\\x^{2} + y^{2} -2xy \geq 0\\(x-y)^{2}\geq 0 (A)$

b)

$x^{2} +x^{2} +x^{2} \geq xy+yx+zy /*2\\\\2x^{2} +2y^{2} +2z^{2} -2xy -2xz -2zy \geq 0$

x²-xy-y²+y²-yz+z²+x²-xz+z²≥0

(x-y)² + (y-z)² + (z-y)²≥0 (A)

c)

xy/z+zx/y+yz/x≥ x+y+z (aducem la acelasi numitor , respectiv xyz, si immultim ambele parti ale inecuatiei cu acesta.)

(xy)² +(yz)² + (zx)² ≥ (x+y+z)xyz

xy notam "a"; yz not "b" zx not "c";

(x+y+z)xyz = x²yz+xy²z+xyz² = xy*xz+ xy*yz + xz*zy =ac+bc+ab

Inlocuim in inecuatie:

a² + b² + c²≥ ac+bc+ab

si de aici se rezolva ca in exemplul b)