Question

Dacă E(x) = 4x : x^2 + 1 , atunci demonstrați că -2 < E(x) < 2 , oricare ar fi x € IR \ { -1 ; 1 }

Answer 1

[tex]E(x)=\frac{4x}{x^2+1}\\ -2\ \textless \ E(x)\ \textless \ 2 (?)\\ .........................\\ -2\ \textless \ E(x)\\ -2\ \textless \ \frac{4x}{x^2+1}\\ -2x^2-2\ \textless \ 4x\\ -x^2-1\ \textless \ 2x\\ x^2+2x+1\ \textgreater \ 0\\ (x+1)^2\ \textgreater \ 0(A)\Rightarrow -2\ \textless \ E(x)\\ ..................................................\\ E(x)\ \textless \ 2\\ \frac{4x}{x^2+1}\ \textless \ 2\\ 4x\ \textless \ 2x^2+2\\ 2x\ \textless \ x^2+1\\ x^2-2x+1\ \textgreater \ 0\\ (x-1)^2\ \textgreater \ 0(A)\Rightarrow E(x)\ \textless \ 2\\ Deci:-2\ \textless \ E(x)\ \textless \ 2 \vee x\in R-\{-1,1\}[/tex]