Dupa efectuarea calculelor, obtinem:
[tex](1+x) ^{5} + (1-x)^{5} =10 x^{4}+20 x^{2} +2.[/tex]
Deci trebuie sa demonstram ca [tex]10x^{4}+20 x^{2} +2 \geq 32. [/tex]
[tex]10 x^{4}+20 x^{2} +2 \geq 32\ \textless \ =\ \textgreater \ \\ \ \textless \ =\ \textgreater \ 10 x^{4}+20 x^{2} \geq 30\ \textless \ =\ \textgreater \ \\ \ \textless \ =\ \textgreater \ x^{4} +2 x^{2} \geq 3\ \textless \ =\ \textgreater \ \\ \ \textless \ =\ \textgreater \ x^{4} + 2x^{2} +1 \geq 4\ \textless \ =\ \textgreater \ \\ \ \textless \ =\ \textgreater \ ( x^{2} +1) ^{2} \geq 4[/tex]
Ramane de demonstrat ca [tex]( x^{2} +1) \geq 4^{2} [/tex].
[tex]|x| \geq 1~=\ \textgreater \ ~ x^{2} \geq 1~=\ \textgreater \ ~ x^{2} +1 \geq 2. \\ \\ Deci~( x^{2} +1) ^{2} \geq 2^{2}=4=\ \textgreater \ ~inegalitate~din~enunt~este~adevarata.[/tex]
