Question

Se consider functiile f; R-> R, f(x) = x^2- 2x+a si g:R ->R, g(x) =-x^2+ 2bx+1, unde
a si b sunt numere reale. Determinati numerele reale a si b, stiind cã parabolele asociate celor douá functii au acelasi vârf.

Answer 1

$V(\frac{-b}{2a} ; \frac{-\Delta}{4a} )$

$f(x)=x^2-2x+a\\\Delta=4-4a\\V(\frac{2}{2} ,\frac{4a-4}{4} )=V(1,a-1)$

$g(x)=-x^2+2bx+1\\\Delta=4b^2-4(-1)=4b^2+4\\V(\frac{-2b}{-2} ,\frac{-4b^2-4}{-4} )=V(b,b^2+1)$

$V(1;a-1)=V(b;b^2+1) \iff \left \{ {{b=1} \atop {a-1=b^2+1}} \right. \\a-1=2 \implies a=3$

$f(x)=x^2-2x+3\\g(x)=-x^2+2x+1$