, 2021

If $x = 1$, then $x^2 = 1$ and $x^3 = 1$ and so $x^3=x^2$.

If $\text{x gt;1}$, then $x^3$ equals $x$ times $x^2$; since $\text{x gt;1}$, then $x$ times $x^2$ is greater than $x^2$ and so $\text{x 3 gt; x 2}$.

Therefore, if $x$ is positive with $\text{x 3 lt; x 2}$, we must have $\text{0 lt;x lt;1}$. We note that if $\text{0 lt; x lt; 1}$, then both $x$, $x^2$ and $x^3$ are all positive, and $\text{x 3 = x 2 x lt; x 2 1 = x 2}$.

Of the given choices, only $x = \frac{3}{4}$ satisfies $\text{0 lt; x lt; 1}$, and so the answer is (B).

We can verify by direct calculation that this is the only correct answer from the given choices.