Answer
$$
f^{\prime \prime}(x)=-2+12 x-12 x^{2} ,\quad f(0)=4 ,\quad f^{\prime}(0)=12
$$
The required function is
$$
f(x)=-x^{2}+2 x^{3}-x^{4}+12 x+4
$$
Work Step by Step
$$
f^{\prime \prime}(x)=-2+12 x-12 x^{2} ,\quad f(0)=4 ,\quad f^{\prime}(0)=12
$$
The general anti-derivative of $
f^{\prime \prime}(x)=-2+12 x-12 x^{2}
$ is
$$
f^{\prime}(x)=-2 x+6 x^{2}-4 x^{3}+C
$$
To determine C we use the fact that $f^{\prime}(0)=12 $:
$$
f^{\prime}(0)=-2 (0)+6(0)^{2}-4 (0)^{3}+C =12
$$
$ \Rightarrow $
$$
0+C=12\quad \Rightarrow \quad C=12,
$$
so
$$
f^{\prime}(x)=-2 x+6 x^{2}-4 x^{3}+12
$$
Using the anti-differentiation rules once more, we find that:
$$
f(x)=-x^{2}+2 x^{3}-x^{4}+12 x+D
$$
To determine D we use the fact that $f(0)=4 $:
$$
f(0)=-(0)^{2}+2 (0)^{3}-(0)^{4}+12 (0)+D =4
$$
$ \Rightarrow $
$$
0+D=4\quad \Rightarrow \quad D=4,
$$
so the required function is
$$
f(x)=-x^{2}+2 x^{3}-x^{4}+12 x+4
$$
