Consider the function f(x) whose second derivative is f''(x)=8x+4sin(x).

Question

If
f(0)=3 and
f'(0)=2, what is f(x)?

f(x)=??

Answer 1

what, no ideas at all? Just straightforward integration.

f"(x) = 8x+4sin x
f'(x) = 4x^2 - 4cos x + C
So, since f'(0) = 2,
16-4+C = 2
C = -10
and so
f'(x) = 4x^2 - 4cos x - 10

Now do that all over again to get f(x)