I have a question that asks:
"Let there be two goods 1 and 2.Let $x$ and $y$ denote their respective quantities.$(x,y)$ represents a bundle. Suppose a consumer’s preferences over bundles in $R^2_+$ can be represented by the utility function $U(x,y)=min(x,2y)$. Prove that the consumer’s preferences over bundles in $R^2_+$ are convex by proving that the utility function is quasiconcave in $R^2_+$"
I was taught that if a utility function is quasiconcave, then $\begin{vmatrix} 0 & U_x & U_y \\ U_x & U_{xx} & U_{xy} \\ U_y & U_{yx} & U_{yy} \\\end{vmatrix} > 0 \forall (x, y) \in R^2_+$
However, when I start doing the partial derivatives, I notice that I get:
$\frac{\partial U}{\partial x} = \begin{cases}1, && x \le 2y \\ 0, && x > 2y\end{cases},$ $\frac{\partial U}{\partial y} = \begin{cases}0, && x \le 2y \\ 2, && x > 2y\end{cases},$
$\frac{\partial U}{\partial x^2} = \begin{cases}0, && x \le 2y \\ 0, && x > 2y\end{cases},$ $\frac{\partial U}{\partial y^2} = \begin{cases}0, && x \le 2y \\ 0, && x > 2y\end{cases},$
$\frac{\partial U}{\partial xy} = \begin{cases}0, && x \le 2y \\ 0, && x > 2y\end{cases},$ $\frac{\partial U}{\partial yx} = \begin{cases}0, && x \le 2y \\ 0, && x > 2y\end{cases}$
I then got $\begin{vmatrix} 0 & U_x & U_y \\ U_x & U_{xx} & U_{xy} \\ U_y & U_{yx} & U_{yy} \\\end{vmatrix} = \begin{cases}0, && x \le 2y \\ 0, && x > 2y\end{cases}$
This is where I am stuck. I am unsure if:
a) I made a mistake somewhere
b) my method is wrong
c) Everything I've done so far is correct, but I need to do another step
Please let me know if I need to provide more information, thanks