Question

find the function f(x) whose second derivaive is f"(x)=12x-sin(x)+1 and satisfies f(0)=-2 and f'(0)=2

Answer 1

first anti derivative is
\(f'(x)=6x^2+\cos(x)+x+c\)
find \(c\) by replacing \(x\) by 0 and setting the result equal to 2

Answer 2

so c is going to be 2?

Answer 3

no i think \(c=1\)

Answer 4

\(f'(0)=\cos(0)+c=1+c=2\) so \(c=1\)

Answer 5

2x^3 + sinx + 0.5x^2 + x - 2

Answer 6

how did you got that @logical reason?

Answer 7

Alright...
As satellite said the first derivative would be
6x^2 + cosx + x +1
Take integral again and you get
2x^3 + sinx + 0.5x^2 +x +c
Substitute x=0 and you get
f(0) = c = -2
Therefore answer is
2x^3 + sinx + 0.5x^2 +x - 2