2022-164

Consider functions \( f \) and \( g \) satisfying:

\( f'' = x^{-1/3} \)
\( g = 4 x^5 + 2 x^3 - 3 x + 5 \)

Which option below is compatible with the preceding equations?

1. \( f''' = -(1/3)x^{-(3/4)} \) and \( g' = 20 x^4 + 6 x - 3 \)
2. \( f''' = -(1/3)x^{-(3/2)} \) and \( g' = 20 x^4 + 6 x^2 - 3 \)
3. \( f''' = (3/2)x^{2/3} \) and \( g' = 4 x^4 + 2 x^2 - 3 \)
4. \( f''' = (3/2)x^{2/3} \) and \( g' = 20 x^4 + 6 x^2 - x \)
5. None of the above

Original idea by: Fábio Assunção.