Question

Cum se calculeaza (x+y)(y+z)(z+x)mai mare sau egal decat 8xyz unde x,y,z apartine de R si sunt numere pozitive

Answer 1

Folosim inegalitatea dintre media aritmetica si media geometrica.
[tex] \frac{x+y}{2} \geq \sqrt{x\cdot y} =>x+y\geq2\sqrt{x \cdot y}\\ \frac{y+z}{2} \geq \sqrt{y\cdot z} =>y+z\geq2\sqrt{y \cdot z}\\ \frac{z+x}{2} \geq \sqrt{z\cdot x} =>z+x\geq2\sqrt{z \cdot x}\\ (x+y)(y+z)(z+x)\geq8\sqrt{x^2\cdot y^2 \cdot z^2}\\ (x+y)(y+z)(z+x)\geq8\cdot x\cdot y \cdot z[/tex]