Question

Stabiliti semnul si monotonia functiilor:
a) f:|R->|R, f(x) = 3x+2
b) f:|R->|R, f(x) =-x+1

Answer 1

a) R(x1,x2)= (f(x2)-f(x1))/(x2-x1)
(3x2+2-3x1-2)/(x2-x1) = (3x2-3x1)/(x2-x1) = (3(x2-x1))/(x2-1)= 3 => f(x) crescatoare
(simplificam cu x2-x1)
b) (-x2+1 + x1-1)/(x2-x1)= (-x2+x1)/(2-x1)= -1 => f(x) descrescatoare
(sam - factor comun sus si ne da -(x2-x1) )

Answer 2

a)
f(x)=0, 3x+2=0, x=-2/3

f(x)>0, x>-2/3

f(x)<0, x<-2/3

-infinit_____-____-2/3__+__0______+____ +infinit_

b)
f(x)=0, -x+1=0, x=1

f(x)>0, x<1

f(x)<0, x>1

-infinit____+____0___+____1______-____________+infinit

f(x)=ax+b

a>0, rezulta f(x)=monoton crescatoare
Deci f(x)=3x+2=monoton crescatoare

a<0, rezulta f(x)=monoton descrescatoare
Deci f(x)=-x+1=monoton descrescatoare