Question

consider the function f(x) whose second derivative is f''(x)=4x+4sin(x). if f(0)=4 and f'(0)=2, what is f(3)?

Answer 1

`\begin{gathered} f^(\prime)(x)=\int 4x+4\sin xdx=2x^2+-4\cos x+c \\ f^(\prime)(0)=-4+c=2\rightarrow c=6 \\ f(x)=\int 2x^2-4cos(x)+6dx=(2)/(3)x^3-4\sin (x)+6x+c \\ f(0)=c=4 \\ f(x)=(2)/(3)x^3-4\sin (x)+6x+4\rightarrow \\ f(3)=18-4\sin (3)+18+4=40-4\sin (3) \end{gathered}`