Question

Aratati ca (a+b)(b+c)(c+a) >_8abc ; a,b,c >_0 numere reale.
>_ ( mai mare sau egal).

Answer 1

folosim inegalitatea mediilor, adica Ma ≥ Mg
$\frac{a+b}{2} \geq \sqrt{ab}$
$\frac{b+c}{2} \geq \sqrt{bc}$
$\frac{c+a}{2} \geq \sqrt{ac}$
Imultim aceste trei inegalitati si obtinem
$\frac{(b+a)(b+c)(c+a)}{2*2*2} \geq \sqrt{ a^{2}* b^{2} * c^{2} }$
(b+a)(b+c)(c+a)≥8abc