(Example 4 Given $0<x<\frac{3}{2}$, find the maximum value of the algebraic expression $x^{2}(3-2 x)$.
Solution: Since $0<x<\frac{3}{2}$, we have $x^{2}(3-2 x) \leqslant\left(\frac{x+x+3-2 x}{3}\right)^{3}=1$.
Therefore, when $x=x=3-2 x$, i.e., $x=1$, the expression $x^{2}(3-2 x)$ reaches its maximum value of 1.
Solution Note: Generally, for several positive numbers, when the product is a constant, the sum has a minimum value; when the sum is a constant, the product has a maximum value. However, it is important to note whether the equality can be achieved, i.e., whether there exists at least one point of extremum.