Let f = yx. What is \[\frac{{{\partial ^2}{\text{f}}}}{{\partial {\text{x}}\partial {\text{y}}}}\] at x = 2, y = 1? A. 0 B. ln 2 C. 1 D. \[\frac{1}{{\ln 2}}\]

The correct answer is $\boxed{0}$.

To find $\frac{{{\partial ^2}{\text{f}}}}{{\partial {\text{x}}\partial {\text{y}}}}$, we can use the following formula:

$$\frac{{{\partial ^2}{\text{f}}}}{{\partial {\text{x}}\partial {\text{y}}}} = \frac{\partial}{\partial y} \left[ \frac{\partial f}{\partial x} \right]$$

In this case, we have $f(x, y) = yx$. Therefore,

$$\frac{\partial f}{\partial x} = y$$

and

$$\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial}{\partial y} \left[ y \right] = 0$$

Plugging in $x = 2$ and $y = 1$, we get

$$\frac{{{\partial ^2}{\text{f}}}}{{\partial {\text{x}}\partial {\text{y}}}} \bigg|_{x = 2, y = 1} = 0$$

Therefore, the correct answer is $\boxed{0}$.

The other options are incorrect because they do not give the correct value of $\frac{{{\partial ^2}{\text{f}}}}{{\partial {\text{x}}\partial {\text{y}}}}$ at $x = 2$ and $y = 1$.