math Analysis, help please.

Answer for the first question.

All maxima and minima have values at cornerpoints.

The cornerpoints are (125, 225), (125, 350), and (250, 100) for values of (e, s).

At e at $6.00 and s at $12.00 per liter, the revenue produced is:

point A:  (125, 225)  =  $3450.00      with s providing $2700.00

point B:  (125, 350)  =  $4950.00      with s providing $4200.00     <---  currently the max

point C:  (250, 100)  =  $2700.00      with s providing $1200.00

 

Increasing the price of e and keeping he price of s constant will never provide an amount for point A that is

larger than the amount for point B.

So, we just need to compare the amount of point B with the amount of point C.

 

Let E be the cost per liter of e.

point B:  amount:  125E + 4200

point C: amount:   250E + 1200

 

When will these two amount be the same?   125E + 4200  =  250E + 1200

                                                                                   3000  =  125E

 --->   E = $24.00 per liter

Below $24.00 per liter, point B is the max; above $24.00 per liter, point C is the max.

Suggestion for the second question:

1)  Keep all the restraints (equations) for the original problem.

2)  Add these restraints:  s >= 250     e <= 200     e <= 300

3)  Graph all the original and the new equations and find their corner points.

4)  Place the values of these corner points into the revenue formula:  revenue  =  6e + 12s

     to find the maximum revenue.