Problem $1.$ Consider cost minimization problem

$mi{n}_{x_1,x_2}c(x_1,x_1)=w_1{x}_1+w_2{x}_2$

$y=f(x_1,x_2)$

Consider the production functions:

a$)f(x_1,x_2)=2x_1^{\frac{1}{2}}+x_2^{\frac{1}{2}}$;

b$)f(x_1,x_2)=2x_1+x_2$;

c$)f(x_1,x_2)=$min$\{2x_1,x_2\}$;

d$)f(x_1,$tilde$(x{)}_2)=x_1^{\frac{1}{2}}$tilde$(x{)}_2^{\frac{1}{2}},$ where the production factor $x_2$ is fixed in the short run at

the level $x_2=$tilde$(x{)}_2.$

For each of these production functions, show the solution graphically $($draw isoquants

and isocosts and show the point or points that minimize the cost$).$ Also, derive a solution

analytically including conditional factor demand and cost function. USE GRAPHS FOR A$-$D