Answer
The value for $x$ for which the center of mass is lowest is $4.169$cm
Work Step by Step
To start, we can set the bottom of the can as the origin. The mass of the liquid with height, $x$, is given by $.354\frac{x}{12}$. The center of mass of both objects is located at $6$cm from the origin for the can, and $\frac{x}{2}$cm from the origin for the liquid.
Using the formula for center of mass, $x_{cm}=\frac{\left(.14\left(6\right)+\frac{x}{2}\left(.354\right)\left(\frac{x}{12}\right)\right)}{.14+.354\left(\frac{x}{12}\right)}$
Taking the derivative of this value with respect to $x$ and setting it equal to $0$, yields a value of $x_{cm}=4.169$ on the closed interval, $[0,12]$.
