Answer
$$f\left( x \right) = {x^2} - 5x + 4$$
Work Step by Step
$$\eqalign{
& f'\left( x \right) = 2x - 5;{\text{ }}f\left( 0 \right) = 4 \cr
& {\text{Calculating the general solution}} \cr
& f\left( x \right) = \int {f'\left( x \right)} dx \cr
& f\left( x \right) = \int {\left( {2x - 5} \right)} dx \cr
& f\left( x \right) = {x^2} - 5x + C \cr
& {\text{Calculating the particular solution for }}f\left( 0 \right) = 4 \cr
& 4 = {\left( 0 \right)^2} - 5\left( 0 \right) + C \cr
& 4 = C \cr
& {\text{The particular solution is}} \cr
& f\left( x \right) = {x^2} - 5x + 4 \cr
& \cr
& {\text{Graphing general solutions for }}C = - 1,{\text{ 2, 4 and the particular}} \cr
& {\text{solution }}f\left( x \right) = {x^2} - 5x + 4 \cr} $$