If f'(x) = 2x^2 - 6x + 7, find f(x) if f(0) = -8.

f'(x) = 2x^2 - 6x + 7  find f(x) if f(0) =
-8.

We are given the first derivative of the function and
we need to determine the function.

Then we know
that:

f(x) = integral of
f'(x).

==> f(x) = intg (2x^2 - 6x + 7)
dx

==> f(x) = intg (2x^2) dx - intg (6x) dx + intg 7
dx

==> f(x) = 2x^3/3 - 6x^2/2 + 7x +
C

==> f(x) = (2/3)x^3 - 3x^2 + 7x +
C

But we are given that f(0) =
-8.

Then we will substitute with x=
0.

==> f(0) =  0 - 0 + 0 + C =
-8

==> C = -8.

Then, we
will substitute with c= -8 into the
function.

==> f(x) = (2/3)x^2 - 3x^2 +
7x - 8