Determine the function f(x) if f(0)=0 and f'(x)=2x+1.

Since the result of the first derivative is a linear
function, we'll consider f(x) as being a quadratic
function.

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

We'll consider the constraint from
enunciation:

f(0) = 0

We'll
substitute x by 0 in the expression of the quadratic:

f(0)
= a*x^2 + b*0 + c

f(0) = c

But
f(0) = 0, so c = 0.

Now, we'll
differentiate f(x):

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

f'(x) = 2ax + b (1)

We'll
impose the other constraint given by enunciation:

f'(x) =
2x + 1 (2)

We'll put (1) =
(2):

2ax + b = 2x + 1

For the
identity to hold, we'll have to impose that the coefficients of x from both sides have
to be equal and the terms that do not contain x from both sides, to be
equal.

2a = 2

a
=
1

and

b
= 1

The expression of the
original function is:

f(x) =
x^2 + x