Question

Find f. f ''(x) = −2 + 24x − 12x2, f(0) = 7, f '(0) = 16

Answer 1

Integrate twice.

Remember the anti-derivative rule for polynomials:
$\int x^n dx = \frac{x^{n+1}}{n+1} + C$

$f'(x) = \int (-2 +24x -12x^2) dx = -2x + 12x^2 -4x^3 + C \\ \\ f'(0) = 16, C = 16 \\ \\ f(x) = \int (16 -2x+12x^2-4x^3) dx = 16x -x^2 + 4x^3 - x^4 + C \\ \\ f(0) = 7, C = 7$

Answer:
$f(x) = 7 + 16x -x^2 + 4x^3 - x^4$