What is the curve which has a slope of 6x^2 at (x, y) and if the curve passes through (4,8)?

the slope of a curve is given by
f'(x).

Therfore f'(x) =
6x^2.

Therefore the curve f(x) = Int f'(x)dx
.

f(x) = Int 6x^2 dx = (1/2+1)6x^(2+1) +C = (6/3) x^3
+C.

f(x) = 2x^3+C.

Since f(x)
passes therough (4,8) ,

f(4) = 8 , or 2*4^3+C = 8. So 
128+C = 8, Or C = 8-128 = -120.

Therefore we rewite  with C
= -120:

f(x) =
2x^3-128.

Therefore the required curve is f(x ) = 2x^3
-120.