Question

Demonstrati ca (x+y)la patrat<=2(x patrat+y patrat) .

Answer 1

(x+y ) ² ≤ 2(x² +y²) 
x² +2xy +y² ≤ 2x² +2y²
0≤ 2x² +2y² -x²-y² -2xy
0 ≤ x² - 2xy +y² 
 ( x- y) ² ≥ 0 adevarat  pentru orice x , y

Answer 2

$(x-y)^2 \geq 0\\ x^2-2xy+y^2 \geq 0|+(x^2+y^2)\\ 2x^2-2xy+2y^2 \geq x^2+y^2\\ 2x^2+2y^2 \geq x^2+2xy+y^2\\ 2(x^2+y^2) \geq (x+y)^2\\ (x+y)^2 \leq 2(x^2+y^2)$