$$max \{ \sqrt{x}+y\}$$
Subject to $$x+2y\le I$$
$$x\ge0$$ $$y\ge 0$$
I want to find the optimal solution by using Kuhn Tucker conditions.
What I do is as follows:
The Lagrangian function
$$L= \sqrt{x}+ y + \lambda [I-x-2y]+\mu_1 [x ] + \mu_2 [y]$$
FOCs
$$\partial L/\partial x= 1/2 x^{-1/2}-\lambda +\mu_1\le 0 $$ $x[1/2 x^{-1/2}-\lambda +\mu_1]=0 $ for $x\ge 0$
$$\partial L/\partial y= 1-2\lambda +\mu_2\le 0 $$
$y[1-2\lambda +\mu_2]=0$ for $y\ge 0$
$$\partial L/\partial\lambda = I- x-2y\ge 0$$
$\lambda [I-x-2y]=0$ for $\lambda \ge 0$
$$\partial L/\partial\mu_1 =x\ge 0$$ $\mu_1 x= 0$ for $\mu_1\ge 0$
$$\partial L/\partial\mu_2 =y\ge 0$$ $\mu_2 y= 0$ for $\mu_2\ge 0$
When I solve this, I found that $(x^*=1, y^*=\frac{I-1}{2}, \lambda=1/2, \mu_2=\mu_1=0)$
I can only found one optimal solution.
But my instructor states there are more than one solutions. I could not proceed the question.
I will be appreciated for your great helps.
Thank you.