Finding the conditional input demand function
Output $z$ is given as $z = x + y$ where $x=min(a,2b)$ and $y = max(3c,4d)$.
So assume that you want $x=12$ then $a=12$ AND $b=6$. Since this part of the production delivers only the minimum of serveral production processes $a$ and $2b$ the producer must get insure that ALL subproductionprocesses $a$ and $2b$ deliver the minimum.
The producer therefore has to buy both $a$ and $b$ in the required amounts and therefore the price for $x$ is composite including both the price of $a$ and $b$.
However, when you want $y=12$ you must have $12=3c$ OR $12=4d$ implying that $c=4$ OR $d=3$. Since the productionprocess $y$ delivers the max of the subproduction processes the producer only has to make sure that AT LEAST ONE of the subproduction processes delivers the wanted amount $y$.
The producer will therefore always produce $y$ using the &quot;cheapest&quot; input and as dependig on input prices. The input price will therefore only depend on the factor $c$ or $d$ chosen to achieve the max.