Question

ofer 100 de puncte.fie f:R->R o functie de gradul I.demonstrati ca are loc relatia f(x-m)+2f(x+2m)=3f(x+m),oricare ar fi m apartine lui R

Answer 1

$O~functie~de~gradul~intai~(liniara)~are~legea~de~forma: \\ \\ f(x)=ax+b. \\ \\ f(x-m)=a(x-m)+b=ax-am+b. \\ \\ f(x+2m)=a(x+2m)+b=ax+2am+b. \\ \\ f(x+m)=a(x+m)+b=ax+am+b. \\ \\ Relatia~de~demonstrat~devine: \\ \\ (ax-am+b)+2(ax+2am+b)=3(ax+am+b).$

$Echivalent~cu: \\ \\ ax-am+b+2ax+4am+2b=3ax+3am+3b \Leftrightarrow \\ \\ \Leftrightarrow 3ax+3am+3b=3ax+3am+3b,~ceea~ce~este~adevarat.$