For the functions $f$ and $g$, is $g(f(A))$ always bigger than $f(g(A))$?

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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?

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

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

3. They are always equal.

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

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

Can you expand or evaluate either of the above expressions?