For the production function in problem $1$A competitive firm has the following production function:

$y=\frac{1}{2}{x}_1+\sqrt[\phantom{2}]{x_2},$

$($a$)$ Find the firm's supply function $y(p,w),$ the factor demand

functions $x_1(p,w)$ and $x_2(p,w),$ and the profit function $(p,w).$

$($b$)$ Verify that $$ is increasing in $p,$ nonincreasing in $w_1,$ homogeneous $1$ in $(p,w),$ and convex in $p($if it is convex in $(p,w),$

then it is also convex in $p).($c$)$ Verify Hotelling$$s Lemma.