Question

F(x)=2×-a/x-b a and b is real number and not = 0 f'(0)=4 and f"(0)=-8 find f(0)

Answer 1

$f(x)=\dfrac{2x-a}{x-b}$
$\implies f'(x)=\dfrac{2(x-b)-(2x-a)}{(x-b)^2}=\dfrac{a-2b}{(x-b)^2}$
$\implies f''(x)=-2\dfrac{2a-2b}{(x-b)^3}=\dfrac{4b-4a}{(x-b)^3}$

$f'(0)=4\implies\dfrac{a-2b}{b^2}=4$
$f''(0)=-8\implies\dfrac{4b-2a}{-b^3}=-8$

$\begin{cases}a-2b=4b^2\\4b-2a=8b^3\end{cases}\implies\begin{cases}a-2b=4b^2\\2b-a=4b^3\end{cases}$
$\implies 4b^2=-4b^3\iff4b^3+4b^2=0\iff b^2(b+1)=0$
$\implies b=0,b=-1$

$b\neq0\implies b=-1$
$\implies a-2b=4b^2\implies a-2(-1)=4(-1)^2\implies a+2=4\implies a=2$

$\implies f(x)=\dfrac{2x-2}{x+1}$
$\implies f(0)=\dfrac{-2}1=-2$