#### Answer

$$g\left( x \right) = {x^7} - {x^4} + 12x + 12$$

#### Work Step by Step

$$\eqalign{
& g'\left( x \right) = 7{x^6} - 4{x^3} + 12 \cr
& g\left( x \right) = \int {g'\left( x \right)} dx \cr
& then \cr
& g\left( x \right) = \int {\left( {7{x^6} - 4{x^3} + 12} \right)} dx \cr
& find{\text{ the general solution}} \cr
& g\left( x \right) = {x^7} - {x^4} + 12x + C \cr
& {\text{using the initial condition }}g\left( 1 \right) = 24 \cr
& 24 = {\left( 1 \right)^7} - {\left( 1 \right)^4} + 12\left( 1 \right) + C \cr
& 24 = 1 - 1 + 12 + C \cr
& C = 12 \cr
& {\text{the solution to the initial value problem is}} \cr
& g\left( x \right) = {x^7} - {x^4} + 12x + 12 \cr} $$