Question

Daca x € R si |x|>=1 sa se demonstreze (1+x)^5+(1-x)^5>=32

Answer 1

Dupa efectuarea calculelor, obtinem:

$(1+x) ^{5} + (1-x)^{5} =10 x^{4}+20 x^{2} +2.$

Deci trebuie sa demonstram ca $10x^{4}+20 x^{2} +2 \geq 32.$

$10 x^{4}+20 x^{2} +2 \geq 32\ \textless \ =\ \textgreater \ \\ \ \textless \ =\ \textgreater \ 10 x^{4}+20 x^{2} \geq 30\ \textless \ =\ \textgreater \ \\ \ \textless \ =\ \textgreater \ x^{4} +2 x^{2} \geq 3\ \textless \ =\ \textgreater \ \\ \ \textless \ =\ \textgreater \ x^{4} + 2x^{2} +1 \geq 4\ \textless \ =\ \textgreater \ \\ \ \textless \ =\ \textgreater \ ( x^{2} +1) ^{2} \geq 4$

Ramane de demonstrat ca $( x^{2} +1) \geq 4^{2}$.

$|x| \geq 1~=\ \textgreater \ ~ x^{2} \geq 1~=\ \textgreater \ ~ x^{2} +1 \geq 2. \\ \\ Deci~( x^{2} +1) ^{2} \geq 2^{2}=4=\ \textgreater \ ~inegalitate~din~enunt~este~adevarata.$