[tex]f(x)=\dfrac{2x-a}{x-b}[/tex]
[tex]\implies f'(x)=\dfrac{2(x-b)-(2x-a)}{(x-b)^2}=\dfrac{a-2b}{(x-b)^2}[/tex]
[tex]\implies f''(x)=-2\dfrac{2a-2b}{(x-b)^3}=\dfrac{4b-4a}{(x-b)^3}[/tex]
[tex]f'(0)=4\implies\dfrac{a-2b}{b^2}=4[/tex]
[tex]f''(0)=-8\implies\dfrac{4b-2a}{-b^3}=-8[/tex]
[tex]\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}[/tex]
[tex]\implies 4b^2=-4b^3\iff4b^3+4b^2=0\iff b^2(b+1)=0[/tex]
[tex]\implies b=0,b=-1[/tex]
[tex]b\neq0\implies b=-1[/tex]
[tex]\implies a-2b=4b^2\implies a-2(-1)=4(-1)^2\implies a+2=4\implies a=2[/tex]
[tex]\implies f(x)=\dfrac{2x-2}{x+1}[/tex]
[tex]\implies f(0)=\dfrac{-2}1=-2[/tex]
