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Determinati intervalul [x:y]⊂R astfel incat (1)[x;y]∩Z=∅ si (2) |y-x-1|=x^2+y^2+x/2-2y+21/16.

Răspuns :

Deoarece [x;y]∩Z=∅ deducem ca intervalul [x,y] nu contine numere intregi=> x si y se gasesc intre doua numere intregi consecutive=> y-x<1=> y-x-1<0=>
|y-x-1|=-(y-x-1)=x-y+1=>
x-y+1=x^2+y^2+x/2-2y+21/16=>
x^2+1/2*x+y^2-y+21/16-1=0=>
(x-1/4)^2+(y-1/2)^2=0=>
x-1/4=0=>x=1/4
y-1/2=0=>y=1/2
[x,y]=[1/4,1/2].