Question

Demonstrati ca (x+y)•(x+y)≤2(x•x+y•y), oricare ar fi x,y .

Answer 1

$( x+y) ^{2} \leq 2( x^{2} + y^{2} )~\Leftrightarrow \\ \Leftrightarrow x^{2} +2xy+ y^{2} \leq 2 x^{2} +2 y^{2} \Leftrightarrow \\ \Leftrightarrow 0 \leq 2 x^{2} +2 y^{2}- x^{2} -2xy- y^{2} \Leftrightarrow \\ \Leftrightarrow 0 \leq x^{2} -2xy+ y^{2} \Leftrightarrow \\ \Leftrightarrow0 \leq (x-y) ^{2} ,care~este~evident~adevart~\Rightarrow~\boxed{(x+y) ^{2} \leq 2( x^{2} + y^{2} )}.$

Answer 2

x=2
y=3
(x+y)·(x+y)=5·5=25 ; 2(x·x+y·y)=2(2·2+3·3)=2(4+9)=2·13=26⇒25≤26⇒
(x+y)≤2(x·x+y·y)