Given

\(f''(x) = 4x - 2\)

and \(f'( -2) =3 \) and \(f( -2)=-3\).

Find \(f'(x) =\) ?

and find \(f( 3) = \) ?

THESHADOW May 3, 2022

#1**+1 **

Note that \(f''(x) = \dfrac{d}{dx} f'(x)\).

Then \(\dfrac{d}{dx} f'(x) = 4x - 2\).

Integrate on both sides, we have \(f'(x) = 2x^2 - 2x + C\) for some real constant C. (Exercise: Try to write out the integration process on your own.)

Since f'(-2) = 3, we know that

\(2(-2)^2 - 2(-2) + C = 3\\ C = -9\)

Then f'(x) = 2x^2 - 2x - 9.

To find f(x), it is pretty much the same process, just with f''(x) replaced with f'(x) and f'(x) replaced with f(x). Please try to do that on your own.

MaxWong May 3, 2022