Question

Fie functia f : R \ {-3) -> R, f(x) = (mx+n)/(x+3). Sa se determine m si n astfel incat punctul A(3,-2) apartine Gf si tangenta in punctul A sa fie inclinata la 45 de grade fata de axa Ox. Multumesc!

Answer 1

functie omografica
monotona

avem conditiile f(3)=-2
si f '(3)=1
pt ca 45de grade cu axa o x face prima bisectoare y=x care are panta 1

f'(x) = (m(x+3) -mx-n) /(x+3)²
f '(x)=(3m-n)/(x+3)²


deci
cele 2 ecuatii  in m si n sunt
f(3)=(3m+n)/6=-2
si
f'(3)=(3m-n)/6²=1


sau 3m+n=-12
3m-n=36
adunam
6m=24
m=4
n=12-3m=-12-12=-24

m=4
n=-24
f(x)=(4x-24)/(x+3)

verificare
f(3)=-12/6=-2 adevarat
f'(x)=(4(x+3)-4x)/(x+3)²=12/(x+3)²
f'(3)=(12-(-24))/36=36/36=1, adevarat,
adevarate ambele, bine rezolvat