Find f(x) if f'(x) = 3x^2 - 5x + 1 if f(0) = 4

Since the first derivative is a quadratic, then the original
function is a polynomial of 3rd order.

We'll put
f(x):

f(x) = ax^3 + bx^2 + cx + d

To
determine f(x), we'll have to calculate the coefficients a,b,c,d.

We
know, from enunciation, that f(0) = 4.

f(0) = d =>
d = 4

Now, we'll differentiate
f(x):

f'(x) = 3ax^2 + 2bx + c

From
enunciation, we know that: f'(x) = 3x^2 - 5x + 1   

We'll
put:

3ax^2 + 2bx + c = 3x^2 - 5x +
1   

We'll put the correspondent coefficients as
equal:

3a = 3

a =
1

2b =
-5

b =
-5/2

c =
1

The original function
is:

f(x) = x^3 - 5x^2/2 + x +
4