Determine the exact mathematical interval(s) in which the function defined by $f(x) = 5x^{3/2} - 3x^{5/2}$ for all $x > 0$ is strictly increasing.
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Always check the initial domain constraints given in the problem statement. Omitting $x > 0$ might lead to incorrectly including negative values or choosing a closed interval like $(-\infty, 1]$.