Question

x,y>= 1
x√y-1 + y√x-1 <= xy

Answer 1

Ma>=Mg inseamna ca s-a folosit inegalitatea dintre media aritmetica si cea geometrica.
Mp>=Ma inseamna ca s-a folosit inegalitatea dintre media patratica si cea aritmetica.

Answer 2

$\displaystyle Voi~demonstra~inegalitatea~utilizand~metoda~spargerii. \\ \\ Inegalitatea~este~echivalenta~cu~\frac{\sqrt{x-1}}{x}+ \frac{\sqrt{y-1}}{y} \le 1. \\ \\ Voi~demonstra~ca~ \frac{\sqrt{a-1}}{a} \le \frac{1}{2}~ \forall~a \ge 1. ~(*)\\ \\ Intr-adevar:~(*) \Leftrightarrow 2 \sqrt{a-1} \le a,~si~cum~a \ge 1 ~si~ 2 \sqrt{a-1} \ge 0, \\ \\ prin~ridicare~la~patrat,~aceasta~devine: 4(a-1) \le a^2 \Leftrightarrow \\ \\ \Leftrightarrow a^2-4a+4 \ge 0 \Leftrightarrow (a-2)^2 \ge0,~adevarat!$

$\displaystyle Deci~ \frac{\sqrt{x-1}}{x}+ \frac{\sqrt{y-1}}{y} \le \frac{1}{2}+ \frac{1}{2}=1,~q.e.d.$