Demand q for product B depends on its price p and its quality (a product attribute) x: q = (a – bp + dx). Profit is q(p – c); the variable cost increases with its quality: c = x2; and a, b and d are positive constants. You wish to choose the price and attribute value that generates maximum profit. Assume there are no constraints (i. e.: any real value of p, x, and q are possible).

a. Write this problem mathematically as a standard optimization problem.
b. Write the first order necessary optimality conditions for this problem (you do not need to solve them explicitly).
c. Is this point a local maximum? Prove it.
d. Is this point a global maximum? Prove it.
e. What happens when you use the starting point [0,20] instead of [0,0]? Why does this happen?

For each of the following cases determine the ending balance in the inventory account. (hint: first, determine the total cost of inventory available for sale. next, subtract the cost of the inventory sold to arrive at the ending balance.)a. jill’s dress shop had a beginning balance in its inventory account of $40,000. during the accounting period jill’s purchased $75,000 of inventory, returned $5,000 of inventory, and obtained $750 of purchases discounts. jill’s incurred $1,000 of transportation-in cost and $600 of transportation-out cost. salaries of sales personnel amounted to $31,000. administrative expenses amounted to $35,600. cost of goods sold amounted to $82,300.b. ken’s bait shop had a beginning balance in its inventory account of $8,000. during the accounting period ken’s purchased $36,900 of inventory, obtained $1,200 of purchases allowances, and received $360 of purchases discounts. sales discounts amounted to $640. ken’s incurred $900 of transportation-in cost and $260 of transportation-out cost. selling and administrative cost amounted to $12,300. cost of goods sold amounted to $33,900.a& b. cost of goods avaliable for sale? ending inventory?