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)

the picture is Thai language.

Answer 1

`f(x)=(2x-a)/(x-b)`

`\implies f'(x)=(2(x-b)-(2x-a))/((x-b)^2)=(a-2b)/((x-b)^2)`

`\implies f''(x)=-2(2a-2b)/((x-b)^3)=(4b-4a)/((x-b)^3)`


`f'(0)=4\implies(a-2b)/(b^2)=4`

`f''(0)=-8\implies(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\\eq0\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)=(2x-2)/(x+1)`

`\implies f(0)=\frac{-2}1=-2`