Answer
$$F\left( x \right) = 2{x^8} + {x^4} + 2x + 1$$
Work Step by Step
$$\eqalign{
& F'''\left( x \right) = 672{x^5} + 24x,{\text{ }}F''\left( 0 \right) = 0,{\text{ }}F'\left( 0 \right) = 2,{\text{ }}F\left( 0 \right){\text{ = 1}} \cr
& F''\left( x \right) = \int {F'''\left( x \right)} dx \cr
& {\text{Substitute }}F'''\left( x \right) \cr
& F''\left( x \right) = \int {\left( {672{x^5} + 24x} \right)dx} \cr
& F''\left( x \right) = 112{x^6} + 12{x^2} + C \cr
& {\text{Use the initial condition }}F''\left( 0 \right) = 0 \cr
& 0 = 112{\left( 0 \right)^6} + 12{\left( 0 \right)^2} + C \cr
& C = 0,{\text{ then}} \cr
& F''\left( x \right) = 112{x^6} + 12{x^2} \cr
& \cr
& F'\left( x \right) = \int {F''\left( x \right)} dx \cr
& {\text{Substitute }}F''\left( x \right) \cr
& F'\left( x \right) = \int {\left( {112{x^6} + 12{x^2}} \right)dx} \cr
& F'\left( x \right) = 16{x^7} + 4{x^3} + C \cr
& {\text{Use the initial condition }}F'\left( 0 \right) = 2 \cr
& 2 = 16{\left( 0 \right)^7} + 4{\left( 0 \right)^3} + C \cr
& C = 2,{\text{ then}} \cr
& F'\left( x \right) = 16{x^7} + 4{x^3} + 2 \cr
& \cr
& F\left( x \right) = \int {F'\left( x \right)} dx \cr
& {\text{Substitute }}F'\left( x \right) \cr
& F\left( x \right) = \int {\left( {16{x^7} + 4{x^3} + 2} \right)dx} \cr
& F\left( x \right) = 2{x^8} + {x^4} + 2x + C \cr
& {\text{Use the initial condition }}F\left( 0 \right) = 1 \cr
& 1 = 2{\left( 0 \right)^8} + {\left( 0 \right)^4} + 2\left( 0 \right) + C \cr
& C = 1,{\text{ then}} \cr
& F\left( x \right) = 2{x^8} + {x^4} + 2x + 1 \cr} $$
