The function is given as,
d dx f( x )=4 x 3 − 3 x 4
We have to calculate the integral of ∫ d dx f( x ) .
Use the formula of ∫ x n dx = x n+1 n+1 +A , where A is constant.
∫ d dx f( x ) = ∫ ( 4 x 3 − 3 x 4 ) dx f( x )=4 ∫ x 3 dx−3 ∫ x −4 dx =4 x 4 4 −3 x −3 −3 +D = x 4 + 1 x 3 +D (1)
It is given that f( 2 )=0,
Substitute the value of x= 2 in equation (1).
f( 2 )= ( 2 ) 4 + 1 ( 2 ) 3 +D =16+ 1 8 +D D=− 129 8
Substitute the value of D in equation (1).
f( x )= x 4 + 1 x 3 − 129 8
Hence, option (A) is correct.