Question

Fie functia f:R-->R, f(x)=(m+1)*x^2-(2m-1)*x+m+1.Sa se determine m apartine R astfel incat :

a) Multimea Gf ∩Ox sa fie formata dintr-un singur punct;

b) Gf intersecteaza Ox in doua puncte distincte;
va rog help me ⊕

Answer 1

$\it card(Gf\cap Ox)=1 \Rightarrow \Delta =0 \Rightarrow (2m-1)^2-4(m+1)^2=0 \Rightarrow \\ \\ \Rightarrow (2m-1)^2-2^2(m+1)^2=0 \Rightarrow (2m-1)^2-(2m+2)^2=0 \Rightarrow \\ \\ \Rightarrow (2m-1-2m-2)(2m-1+2m+2)=0 \Rightarrow -3(4m+1)=0 \Rightarrow \\ \\ \Rightarrow 4m+1=0 \Rightarrow 4m=-1 \Rightarrow m=-\dfrac{1}{4}$

$\it b)\ card(Gf\cap Ox)=2 \Rightarrow \Delta>0 \Rightarrow -3(4m+1)>0 \Rightarrow 4m+1<0 \Rightarrow \\ \\ \Rightarrow 4m<-1 \Rightarrow m<-\dfrac{1}{4} \Rightarrow m\in(-\infty,\ \ -\dfrac{1}{4})$