Question

11.Daca x este un numar real pozitiv,aratati ca $x+ x^{-1}$≥2.
12.Daca $\sqrt{186abc}$∈N,a≠0,b≠0,c≠0,aflati media geometrica a numerelor x=a+b+c si y=a*b*c

Answer 1

[tex]x+x^{-1} \geq2 \Longleftrightarrow x+\dfrac{1}{x} \geq2 \Longleftrightarrow x^2+1 \geq 2x \Longleftrightarrow x^2-2x+1 \geq 0 \\\;\\ \Longleftrightarrow (x-1)^2\geq0 (Adevarat)[/tex]

12) 

[tex]431\ \textless \ \sqrt{\overline{186abc}}\ \textless \ 433 \Rightarrow \overline{186abc} = 432^2=186\ 624 \Rightarrow a=6,\ b=2,\ c=4 \\\;\\ x=a+b+c=6+2+4=12 \\\;\\ y =a\cdot b\cdot c =6\cdot2\cdot4=48. \\\;\\ m_g=\sqrt{x\cdot y} =\sqrt{12\cdot48}=\sqrt{12\cdot12\cdot4} =\sqrt{144\cdot4} =12\cdot2=24. \\\;\\ .[/tex]