Let

\[f(x) = \int_0^1(xt)^2\:dt, \quad \text{and} \quad g(x) = \int_0^xt^2\:dt.\]

Let \(A>0\). Which of the following statements is true?

\(g(f(A))\) is always bigger than \(f(g(A))\).

\(f(g(A))\) is always bigger than \(g(f(A))\).

They are always equal.

\(f(g(A))\) is bigger if \(A < 1\), and \(g(f(A))\) is bigger if \(A > 1\).

\(g(f(A))\) is bigger if \(A < 1\), and \(f(g(A))\) is bigger if \(A > 1\).

Can you expand or evaluate either of the above expressions?