Question $5$

Solve each of the following optimization problems, Don't forget to set up the lagrangian before solving for the optimal points.

$($a$)$

$ma{x}_{x,y}u(x,y)=xys.t.2x+y=100.$

$($b$)$

$mi{n}_{K,L}K+20Ls.t.\sqrt[\phantom{2}]{K}+L=30.$

$($c$)$

$ma{x}_{x,y}u(x,y)=\sqrt[\phantom{2}]{x}+3,s.t.,x+4y=100$