Inegalitate conditionata de modul

[Antonache Emanuel]

Fie \( a\in\mathbb{R} \), \( |a|\leq4 \). Demonstrati ca:
\( x^4+y^4+axy+2\geq 0 \),\( \forall x, y\in\mathbb{R} \)

[BogdanCNFB]

\( |a|\le 4\Rightarrow -4\le a\le 4\Rightarrow a\le -4 \)
Avem \( x^4+y^4+axy+2\ge x^4+y^4-4xy+2=x^4+y^4-2x^2y^2+2x^2y^2-4xy+2=(x^2-y^2)^2+2(xy-1)^2\ge 0 \).

[Marius Mainea]

\( x^4+y^4+axy+2\ge x^4+y^4-4xy+2 \)

Nu prea e adevarat

\( |axy|\le\frac{|a|(x^2+y^2)}{2}\le 2(x^2+y^2) \) si de aici

\( axy\ge-2(x^2+y^2) \)

Apoi inegalitatea este evidenta:

\( LHS\ge x^4+y^4-2(x^2+y^2)+2=(x^2-1)^2+(y^2-1)^2\ge 0 \)