Maximum of a fractional power product

Find the maximum value of $x^{\frac{1}{2}} (1 - x)^{\frac{1}{3}}$ when $x$ is a proper fraction.

~10 minAnswer checked instantlyInequalitiesMaxima and minima

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XIX. Inequalities, Examples XIX.b, No. 10 (part 2)

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Checked against the answer in H. S. Hall & S. R. Knight, Higher Algebra (1891).