Determine the function fx if f00 and fx2x1SolutionSince the

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

Solution

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