Let $x_{ij}$ denote the amount of good $i$ bought by the consumer from store $j$. So, consumer's utility maximization problem is described as follows :
\begin{eqnarray*} \max_{x_{11}, x_{12}, x_{21}, x_{22}} & (x_{11}+x_{12})(x_{21}+x_{22}) \\ \text{s.t.} & \ \ 2\sqrt{x_{11}} + x_{21} + 2x_{12} + \frac{1}{2} x_{22} \leq 6 \\ & x_{11} \geq 0, x_{12} \geq 0, x_{21} \geq 0, x_{22} \geq 0 \end{eqnarray*}
Given that good 2 is cheaper in store 2, we know that the solution to this problem will satisfy $x_{21} = 0$. Consequently, it is clear from the objective that consumer will always choose $x_{22} > 0$. Also consumer's utility is an increasing function, so consumer will spend all his income in optimum. Taking them into account, we can rewrite the utility maximization problem of the consumer as :
\begin{eqnarray*} \max_{x_{11}, x_{12}, x_{22}} & (x_{11}+x_{12})x_{22} \\ \text{s.t.} & \ \ 2\sqrt{x_{11}} + 2x_{12} + \frac{1}{2} x_{22} = 6 \\ & x_{11} \geq 0, x_{12} \geq 0, x_{22} > 0\end{eqnarray*}
Now we set up the Lagrangian:
\begin{eqnarray*} \mathcal{L}(x_{11}, x_{12}, x_{22}) = (x_{11}+x_{12})x_{22} - \lambda (2\sqrt{x_{11}} + 2x_{12} + \frac{1}{2} x_{22} - 6) + \mu_1 x_{11} + \mu_2 x_{12} \end{eqnarray*}
Corresponding Kuhn-Tucker conditions are :
\begin{eqnarray*} \frac{\partial \mathcal{L}}{\partial x_{11}} & = & x_{22} - \frac{\lambda}{\sqrt{x_{11}}} + \mu_1 = 0 \\ \frac{\partial \mathcal{L}}{\partial x_{12}} & = & x_{22} - 2\lambda + \mu_2 = 0 \\ \frac{\partial \mathcal{L}}{\partial x_{22}} & = & x_{11} + x_{12} - \frac{\lambda}{2} = 0 \\ \frac{\partial \mathcal{L}}{\partial \lambda} & = & 6 - 2\sqrt{x_{11}} - 2x_{12} - \frac{1}{2} x_{22} = 0, \ \lambda \geq 0 \\ \frac{\partial \mathcal{L}}{\partial \mu_1} & = & x_{11} \geq 0, \ \mu_1 \geq 0, \ \mu_1x_{11} = 0 \\ \frac{\partial \mathcal{L}}{\partial \mu_2} & = & x_{12} \geq 0, \ \mu_2 \geq 0, \ \mu_2x_{12} = 0\end{eqnarray*}
One of the solutions to the above conditions solve the optimization problem. Here is that solution :
$x_{11}^* = 4, x_{12}^* = 0, x_{22}^* = 4, \lambda^* = 8, \mu_1^* = 0,\mu_2^* = 12$.
Therefore, consumer will buy 4 units of commodity 1 and 0 unit of commodity 2 from the first store; and 0 unit of commodity 1 and 4 units of commodity 2 from the other store.
