Notes

First up, you're going to want to expand and simplify the revenue function. To do that first expand your bracket.



(x-120)2 = (x-120)(x-120)
Expand using FOIL. You will get (x2 - 120x - 120x + 14400)
Simplify to get (x2 - 240x + 14400)
Lets put that expanded form back in our equation:
-0.5 (x2 - 240x + 14400) + 7200
Now multiply out the bracket by -0.5
-0.5x2 + 120x -7200 + 7200
Note that the -7200 +7200 cancel to make 0
We end up with our revenue function being: -0.5x2 + 120x
To find the profit function we need to do the revenue function subtract the cost function.
-0.5x2 + 120x - (40x + 200)
We expand to get: -0.5x2 + 120x - 40x - 200
Now simplify: -0.5x2 + 80x - 200
This is our profit function.


For the Domain



The question tells us the most we items (x) we can make is 160. We can't make negative items so the least is 0.

This can be written as 0 ≤ x ≤ 160 or in interval notation [0,160]



For the final components of the question we need to sub 80 and 90 into our x's in our profit function.



-0.5(80)2 + 80(80) - 200
3000
-0.5(90)2 + 80(90) - 200
2950
From those two levels of production the greatest profit is achieved when 80 items are produced, so this is the level of production they should opt for.
For the last stage, graphing the equation can help but you can also do a quick check by subbing in x=79 and x=81 to see that x is the maximum point of the graph, meaning it yields the highest possible profit. Also note that the leading coefficient is negative so the x2 curve increases first then decreases (like an upside down u).



I hope that answers your questions!



For the last part (Part 4), graphing the equation will really help but I used a quick check by subbing in x=80 and x=90 to see that x is the maximum point of the graph, meaning it yields the highest possible profit. The leading coefficient is negative so the x2 curve increases first then decreases (like an upside down u).

When a function reaches the optimal value, the value of the function begins to reduce. This means that, producing 10 more units takes the profit function beyond its maximum point.