Question

fie a = radical din 2 -1 b= radical din 2 +1.Aratati ca 1/a +1/b apartine (2,3)

Answer 1

[tex]a=\sqrt2-1\\ b=\sqrt2+1\\ \frac{1}{a}+\frac{1}{b}=\frac{1}{\sqrt2-1}+\frac{1}{\sqrt2+1}=\frac{\sqrt2+1}{(\sqrt2-1)(\sqrt2+1)}+\frac{\sqrt2-1}{(\sqrt2+1)(\sqrt2-1)}=\\ =\sqrt2+1+\sqrt2-1=2\sqrt2\\ Stim\ ca:\\ 1\ \textless \ \sqrt2\ \textless \ 1,5\\ Deci:\\ 2\ \textless \ 2\sqrt2\ \textless \ 3\\ Adica: 2\ \textless \ \frac{1}{a}+\frac{1}{b}\ \textless \ 3\Rightarrow\frac{1}{a}+\frac{1}{b}\in (2,3)[/tex]