Question

daca a > 0 stabileste valoarea de adevar a propoziti A+ \frac{1}{a} \geq 2

Answer 1

$a\ \textgreater \ 0 \\ \\ a+\dfrac{1}{a}\geq2 \\ \\ Conform inegalitatii mediilor: \quad \dfrac{x+y}{2} \geq \sqrt{x\cdot y},\quad \forall x,y \ \textgreater \ 0 \\ \\ Noi avem: \quad \dfrac{a+\dfrac{1}{a}}{2}\geq \sqrt{a\cdot \dfrac{1}{a}} \Rightarrow a+\dfrac{1}{a}\geq 2\cdot \sqrt{1}\Rightarrow \boxed{a+\dfrac{1}{a}\geq2} (A)$