+136 Given
\(f''(x) = 4x - 2\)
and \(f'( -2) =3 \) and \(f( -2)=-3\).
Find \(f'(x) =\) ?
and find \(f( 3) = \) ?
+9673 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.
