 Hello and welcome to the session. Let us discuss the following question. Question says, choose the correct answer. If d by dx of fx is equal to 4x cube minus 3 upon x raised to the power 4 such that f2 is equal to 0, then fx is x raised to the power 4 plus 1 upon x cube minus 1 to 9 upon 8 bx cube plus 1 upon x raised to the power 4 plus 1 to 9 upon 8 c x raised to the power 4 plus 1 upon x cube plus 1 to 9 upon 8 dx cube plus 1 upon x raised to the power 4 minus 1 to 9 upon 8. We have to choose the correct answer from a, b, c and d. Let us now start with the solution. We are given d by dx of fx is equal to 4x cube minus 3 upon x raised to the power 4. Clearly we can see derivative of fx is equal to 4x cube minus 3 upon x raised to the power 4. So this implies anti-derivative or we can say integral of 4x cube minus 3 upon x raised to the power 4 with respect to x is equal to fx. First of all let us find out this integral. Now we know this integral can be written as 4 multiplied by integral of x cube with respect to x minus 3 multiplied by integral of 1 upon x raised to the power 4 with respect to x. Now we will write this integral as it is and we know 1 upon x raised to the power 4 can be written as x raised to the power minus 4. So here we will write 3 multiplied by integral of x raised to the power minus 4 dx. Now we know integral of x raised to the power n with respect to x is equal to x raised to the power n plus 1 upon n plus 1 plus c where c is the constant of integration. Now we will find integral of this term by using this formula. It is equal to 4 multiplied by x raised to the power 3 plus 1 upon 3 plus 1. Clearly we can say value of n is equal to 3 here. So we will get integral of x cube is equal to x raised to the power 3 plus 1 upon 3 plus 1. Now we will find integral of this term here the value of n is equal to minus 4. So we will write it as 3 multiplied by x raised to the power minus 4 plus 1 upon minus 4 plus 1 plus c. C is the constant of integration. Now simplifying this expression further we get 4 multiplied by x raised to the power 4 upon 4 minus 3 multiplied by x raised to the power minus 3 upon minus 3 plus c. 4 and 4 will get cancelled and this 3 and this 3 will get cancelled and we get x raised to the power 4 minus minus x raised to the power minus 3 plus c. Now simplifying further we get x raised to the power 4 plus x raised to the power minus 3 plus c. Now x raised to the power minus 3 can be written as 1 upon x cube. So we will write it as x raised to the power 4 plus 1 upon x cube plus c. So we get integral of 4x cube minus 3 upon x raised to the power 4 with respect to x is equal to x raised to the power 4 plus 1 upon x cube plus c. We also know that that this integral is equal to fx. We have already shown it above. Let us name this expression as 1. So we can write fx is equal to x raised to the power 4 plus 1 upon x cube plus c. We are given in the question that f2 is equal to 0. Now substituting 2 for x in this expression we get f2 is equal to 2 raised to the power 4 plus 1 upon 2 cube plus c. Now this implies f2 is equal to 16 plus 1 upon 8 plus c. Now this further implies f2 is equal to 1 to 9 upon 8 plus c. Adding these two terms we get 1 to 9 upon 8. We are given that f2 is equal to 0. Now substituting this value of f2 in this expression we get 0 is equal to 1 to 9 upon 8 plus c. Now subtracting 1 to 9 upon 8 from both the sides of this expression we get minus 1 to 9 upon 8 is equal to c. Or we can simply write it as c is equal to minus 1 to 9 upon 8. Now we know fx is equal to x raised to the power 4 plus 1 upon x cube plus c. Let us name this expression as 2. Now substituting this value of c in expression 2 we get fx is equal to x raised to the power 4 plus 1 upon x cube minus 1 to 9 upon 8. So our correct answer is a. So this is our required answer. This completes the session. Hope you understood the solution. Take care and have a nice day.