Question

aratati ca (a+b)/c + (b+c)/a + (c+a)/b >= 6, oricare ar fi x din (0, +infinit)

Answer 1

Răspuns:

$\frac{a+b}{c} +\frac{b+c}{a} +\frac{c+a}{a} \geq 6$

$\frac{a}{c} +\frac{b}{c} +\frac{b}{a} +\frac{c}{a} +\frac{c}{b} +\frac{a}{b} \geq 6$

$(\frac{a}{c} +\frac{c}{a} )+(\frac{b}{c} +\frac{c}{b} )+(\frac{a}{b} +\frac{b}{a} )\geq 6$

Anaalizam prima paranteza si observam ca-a maai mare decat 2

$\frac{a}{c} +\frac{c}{a} \geq 2$

duci la acelasi numitor

$\frac{a^2}{ac} +\frac{c^2}{ac} \geq 2\frac{ac}{ac}$

a²+c²≥2ac

a²-2ac+c²≥0

(a-c)²≥0 evident

Analog demonstrezi c si elelalte  paranteze sunt mai mari sau egale cu 2

Deci 2+2+2≥6 Evident

a,b,c>0

Explicație pas cu pas: