I'm trying to find boundary solutions for the following utility maximization problem, but i'm unsure on how to proceed. Here is the problem and what I got so far:
$ \max x_1^\alpha + x_2 \qquad \text{s.t.}\ x_1 \geq 0,\ x_2 \geq 0,\ p_1x_1 + p_2x_2 \leq w$
where $\alpha \in [0,1]$. The Kuhn-Tucker conditions are
$\begin{equation} \dfrac{\partial \mathcal{L}}{\partial x_1} = \alpha x_1 ^{\alpha -1} - \lambda p_1 + \mu_1=0,\\ \dfrac{\partial \mathcal{L}}{\partial x_2} = 1 - \lambda p_2 + \mu_2 =0,\\ p_1x_1 + p_2x_2 \leq w,\\ x_1\geq 0,\\ x_2 \geq 0,\\ \lambda(p_1x_1 + p_2x_2-w)=0,\\ \mu_1x_1 = 0,\\ \mu_2x_2=0,\\ \lambda \geq 0, \quad \mu_1 \geq 0, \quad \mu_2 \geq 0.\\ \end{equation}$
If I consider boundary solutions with $x_1=0$ and $x_2>0$, then by the complementary slackness condition we have $\mu_1 \geq 0$ and $\mu_2=0$.
Since $\nabla u(x_1,x_2) \gg 0$ and $p_1,p_2 > 0$, $\lambda > 0$. The constraint on the budget set is binding (Walras' law holds). My candidate for the optimum therefore is $x_2=w/p_2$.
The Kuhn-Tucker necessary (and in this case also sufficient) conditions for $x_2=w/p_2$ being optimal restrict to
$\begin{equation} \alpha x_1 ^{\alpha -1} \leq \lambda p_1 ,\\ 1 = \lambda p_2,\\ x_1 = 0,\\ x_2 \geq 0. \end{equation}$
By dividing the first condition by the second we obtain
$MRS_{1,2}(x_1,x_2)= \alpha x_1 ^{\alpha -1} \leq \dfrac{p_1}{p_2}$
but then I do not know how to proceed.