E:16902

E:16902 (Melania-Iulia Dobrican)

Fie numerele reale pozitive Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} , cu Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xy=4} . Arătaţi că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{x+4} + \frac{1}{y+4} \le \frac{1}{3}.}

Soluție

Avem echivalenţele ${\frac {1}{x+4}}+{\frac {1}{y+4}}\leq {\frac {1}{3}}\Leftrightarrow 3y+12+3x+12\leq xy+4x+4y+16\Leftrightarrow x+y\geq 8-xy.$ Cum $xy=4$, rezultă $x+y\geq 4$. Au loc echivalenţele Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x+y \ge 4 \Leftrightarrow \frac{x+y}{2} \ge 2=\sqrt{4}\Leftrightarrow \frac{x+y}{2} \ge \sqrt{xy},} inegalitate care are loc pentru orice numere reale pozitive Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} .

Cazul de egalitate are loc pentru Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=y =2} .