Question

Sa se determine intervalul [a, b] stiind ca sunt indplinite simultan conditiile:
a) [a, b] ∩ Z = ∅;
b) |b - a - 1| = a² + b² + a/2 - 2b + 21/16

Answer 1

Răspuns:

[a;b]∩Z=∅⇒a<b<a+1

atunci |b-a-1|=a+1-b

a+1-b=a²+b²+a/2-2b+21/16

a²-a/2+b²-b+5/16=0

a²-a/2+1/16+b²-b+1/4=0

(a-1/4)² +(b-1/2)²=0

cum c²≥0, ∀c∈R

relatia poate fi valabila numai pt.

a-1/4 =0

si

b-1/2=0

deci a=1/4 si b=1/2

verificare

[1/4;1/2]∩Z=∅, adevarat

|1/2-1/4-1|=1/16+1/4+1/8-1+21/16

|-3/4|=5/16+2/16-16/16+21/16

3/4=(28-16)/16

3/4=12/16

3/4=3/4

adevarat

adevarate ambele, bine rezolvat