Answer
$$
f^{\prime }(x)=5x^{4}-3x^{2}+4 , \quad f(-1)=2
$$
The required function is
$$
f(x)=x^{5}-x^{3}+4x +6
$$
Work Step by Step
$$
f^{\prime }(x)=5x^{4}-3x^{2}+4 , \quad f(-1)=2
$$
The general anti-derivative of $
5x^{4}-3x^{2}+4
$ is
$$
f(x)=x^{5}-x^{3}+4x +C
$$
To determine C we use the fact that $f(-1)=2$:
$$
f(x)=(-1)^{5}-(-1)^{3}+4(-1) +C =2
$$
$ \Rightarrow $
$$
-4+C=2 \quad \Rightarrow \quad C=6,
$$
so
$$
f(x)=x^{5}-x^{3}+4x +6
$$
