Chapter 12. Unconstrained Optimization

Pre-Test Question:

Unconstrained Optimization
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You must read each slide, and complete any questions on the slide, in sequence.

Question

Suppose you own a small woodworking company that makes bookshelves. When you make x bookshelves your profit is: π(x)=200x-(150x+5x2). To get the most profit, how many bookshelves should you make?

Numerical free answer space: DYU2tVvtzEQ=

Correct! Making five bookshelves maximizes your profits.
Sorry, Making five bookshelves gives you the most possible profit. The first order condition of the profit function is 50-10x=0 which is solved by x=5. This is a maximum since the second derivative is -10 < 0.

Video

Post-Test Question

Question Post-Test Question 1:

Suppose you are at an all-you-can-eat buffet and are trying to decide how many plates of food to eat; too few and you’ll still be hungry, too many and you’ll be sick. If your utility from eating x plates of food is: U(x)=60x-10x2. How many plates of food should you eat to maximize your utility?

Numerical free answer space: 607M7xmPORU=

Correct! Eating three plates of food will give you the most utility possible.
Sorry, eating three plates of food gives you the most utility possible. The first order condition of the utility function is 60-20x=0 which is solved by x=3. This is a maximum since the second derivative is -20 < 0.

Question Post-Test Question 2:

Suppose you own an internet company that designs websites. If you design x websites your profit will be: π(x)=-21+54x-9x2. To get the most profit, how many websites should you design to maximize your profit?

Numerical free answer space: 607M7xmPORU=

Correct! To maximize your profits you should design three websites.
Sorry, Designing three websites gives you the most possible profit. The first order condition of the profit function is 54-18x=0 which is solved by x=3. This is a maximum since the second derivative is -18 < 0.

Question Post-Test Question 3:

You are having a party at your house and are trying to decide how many guests to invite. If you invite too few it will be boring. If you invite too many it will be too crowded. If you invite x guests the “success” of your party can be measured by f(x)=1000-5x2+200x. How many guests should you invite to maximize the success of your party?

Numerical free answer space: Tu9IG1n3UyE=

Correct! To maximize the success of your party you should invite 20 guests.
Sorry, to maximize the success of your party you should invite 20 guests. The first order condition of the success function is -10x+200=0 which is solved by x=20. This is a maximum since the second derivative is -10 < 0.

Question Post-Test Question 4:

Suppose you own a coffee shop and are trying to decide how many workers to hire. Your coffee shop usually sells 200 cups per day. If you hire x workers to produce this amount of coffee then the cost of production will be: C(x)=15x2-60x. How many workers should you hire to minimize the cost of producing 200 cups per day?

Numerical free answer space: XvVM00l89Is=

Correct! To minimize the cost of production you should hire two workers.
Sorry, to minimize the cost of production you should hire two workers. The first order condition of the cost function is 30x-60=0 which is solved by x=2. This is a minimum since the second derivative is 30 > 0.

Question Post-Test Question 5:

Suppose it’s the weekend, you have an Economics exam on Monday and you’re trying to decide how many hours to study. The more you study the better you’ll do on the exam. The less you study the more free time you’ll have for other activities. If your total utility from studying for x hours is given by u(x)=3x-0.25x2 then how many hours should you study to maximize your total utility?

Numerical free answer space: yBhAQ+3VvjM=

Correct! Studying for six hours will give you the most utility possible.
Sorry, studying for six hours will give you the most utility possible. The first order condition of the utility function is 3-0.5x=0 which is solved by x=6. This is a maximum since the second derivative is -0.5 < 0.