The respective graphs are
For material 2:
and for material 3:
We can test each set of functions by substituying a value of x. For instance, when x= 100 we have
Material 1:
[tex]\begin{gathered} R(100)=200000(100)-2000(100)^2 \\ R(100)=0 \end{gathered}[/tex]
and
[tex]\begin{gathered} c(100)=5000000-20000(100) \\ c(100)=3000,000 \end{gathered}[/tex]
Material 2:
[tex]\begin{gathered} R(100)=160000(100)-1000(100^2) \\ R(100)=6,000,000 \end{gathered}[/tex]
and
[tex]\begin{gathered} C(100)=4000000-10000(100) \\ C(100)=3,000,000 \end{gathered}[/tex]
Material 3:
[tex]\begin{gathered} R(100)=54000(100)-270(100^2) \\ R(100)=27,000,000 \end{gathered}[/tex]
and
[tex]\begin{gathered} C(100)=2000000-5000(100) \\ C(100)=1,500,000 \end{gathered}[/tex]
By comparing results, we can see that material 3 can deliver the highest profit about 27,000,000-1,500,000= 25000000.