Question

Daca x,y ∈[5,7] ,atunci aratati ca xy-6(x+y)+42 ∈ [5,7]

Answer 1

$\displaystyle Avem~xy-6(x+y)+42=xy-6(x+y)+36+6= \\ \\ =(x-6)(y-6)+6. \\ \\ Deci~trebuie~sa~demonstram~ca~(x-6)(y-6) +6 \in [5,7]. \\ \\ Stim~din~enunt~ca~x,y \in [5,7]. \\ \\ Deci~x-6 \in [-1,1]~si~y-6 \in [-1,1]. \\ \\ Rezulta~(x-6)(y-6) \in [-1,1]. \\ \\ Deci~(x-6)(y-6)+6 \in [5,7],~q.e.d.$