[tex]f(x)=\sqrt{x^2+x+1}-\sqrt{x^2+x-1}=\frac{x^2+x+1-x^2-x+1}{\sqrt{x^2+x+1}+\sqrt{x^2+x-1}}=\\
= \frac{2}{\sqrt{x^2+x+1}+\sqrt{x^2+x-1}} >0=>f(x)>0\\
f(x)=\frac{2}{\sqrt{(x+\frac{1}{2})^+\frac{3}{4}}+\sqrt{((x+\frac{1}{2})^2-\frac{5}{4}}} [/tex]
Acest raport ia valoarea maxima daca numitorul e cat mai mic. Al doilea radical de jos ia valoarea cea mai mica 0 pentru [tex](x+\frac{1}{2})^2=\frac{5}{4}=>x+\frac{1}{2}=\pm\frac{\sqrt{5}}{2}[/tex]
Inlocuind [tex]x+\frac{1}{2}=\pm\frac{\sqrt{5}}{2}[/tex] in primul radical se obtine
[tex]\sqrt{(x+\frac{1}{2})^2+\frac{3}{4}}=\sqrt{\frac{5}{4}+\frac{3}{4}}=\sqrt{2}[/tex]
[tex]0<f(x)<\frac{2}{\sqrt{2}}[/tex]
Multimea valorilor functiei f este intervalul :[tex](0;\sqrt{2})[/tex].
