I am reviewing old midterms to prepare for my upcoming midterm and ran across this question:
Let $\alpha , \beta \in (0,1)$. Now, let $f_{\alpha}$ and $f_{\beta}$ on $\mathbb{R^2}$ be defined as $f_{\alpha}(x)=x_1^{\alpha} x_2^{1 - \alpha}$ and $f_{\beta}(x)=x_1^{\beta} x_2^{1 - \beta}$
Now, let R be a binary relation on $\mathbb{R_x^2}$. Let ${x,y} \subset R_+^2$ We have that: $$xRy \leftrightarrow f_{\alpha}(x) \geq f_{\alpha}(y) \land f_{\beta}(x) \geq f_{\beta}(y)$$
For which combinations of $\alpha$ and $\beta$ is this binary relation complete, for which combinations is it transitive and for which combinations is it continuous.
Clarifying the context
This is a question dealing with the 'link' between consumer preferences and utility theory in economics. So $xRy$ means that 'good x is at least as good as good y'. This bridge between the binary relation and the $\geq and \leq$ operations in $\mathbb{R}$ helps bridge preference theory and utility theory.
My thoughts
It seems like $x_1=x_2=y_1=y_2 \implies$ completeness for all combinations of $\alpha$ and $\beta$ and it seems true in a trivial way that $\alpha = \beta$ makes this complete for all vector $x$ and $y$.
I think this is transitive for all combinations of $\alpha$ and $\beta$ where we have $xRy$ and $yRz$ because of the well-ordering of $\mathbb{R_+^2}$
i am not sure how to address continuity.