If f'(x) = 3x^2 - 6x +3, find f(x) if f(0) = 2

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have the derivative of the function f(x) given as f'(x) = 3x^2 - 6x + 3. f(0) = 2

To find f(x), take the integral of f'(x)

Int[ f'(x) dx] = Int[ 3x^2 - 6x + 3 dx]

=> 3*x^3/ 3 - 6*x^2 / 2 + 3x + C

=> x^3 - 3x^2 + 3x + C

As f(0) = 2

0^3 - 3*0^2 + 3*0 + C = 2

=> C = 2

The required function f(x) = x^3 - 3x^2 + 3x + 2

 

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hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

Given that f'(x) = 3x^2 - 6x + 3

We need to find the function f(x).

But we know that f(x) = Integral f'(x).

==> f(x) = Int (3x^2 - 6x + 3) dx

==> f(x) = Int (3x^2) dx - Int 6x dx + Int 3 dx

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

Let us simplify.

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

But we know that f(0) = 2

==> f(0) = 0 - 0 + 0 + C = 2

==> C = 2

==> f(x) = x^3 - 3x^2 +3x + 2

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