Question

sa se demonstreze inegalitatea​

Answer 1

$\it (a+b+c)(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c})=1+\dfrac{a}{b}+\dfrac{a}{c}+\dfrac{b}{a}+1+\dfrac{b}{c}+\dfrac{c}{a}+\dfrac{c}{b}++1=\\ \\ \\ =3+(\underbrace{\dfrac{a}{b}+\dfrac{b}{a}}_{\geq2})+(\underbrace{\dfrac{b}{c}+\dfrac{c}{b}}_{\geq2})+(\underbrace{\dfrac{c}{a}+\dfrac{a}{c}}_{\geq2})\geq3+2+2+2=9,\ \forall a, b, c\in(0,\ \infty)$