[tex] \frac{a}{b}+ \frac{b}{a} \geq 2 <=> \\ <=> \frac{ a^{2} + b^{2} }{ab} \geq 2<=> \\ <=> a^{2} + b^{2} \geq 2ab <=> \\ <=> a^{2}+ b^{2} -2ab \geq 0 <=> \\ <=> (a-b)^{2} \geq 0[/tex]
Ceea ce este evident adevarat => [tex] \frac{a}{b}+ \frac{b}{a} \geq 2 [/tex] (egalitatea are loc cand a=b).
