Maximize the function 𝑓(𝑥,𝑦)=3𝑥𝑦−8𝑥2−4𝑥−5𝑦+12 subject to the constraint 3𝑥+2𝑦=5 Find the location of the maximum and its value. Enter non-integer numerical values as decimals to at least 3 decimal places. Note: you must use a . and not , for a decimal point. The maximum is located at 𝑥= Answer 1 Question 7 , 𝑦= Answer 2 Question 7 , and 𝑓(𝑥,𝑦)= Answer 3 Question 7

Maximize the function 𝑓(𝑥,𝑦)=3𝑥𝑦−8𝑥2−4𝑥−5𝑦+12 subject to the constraint 3𝑥+2𝑦=5 Find the location of the maximum and its value. Enter non-integer numerical values as decimals to at least 3 decimal places. Note: you must use a . and not , for a decimal point. The maximum is located at 𝑥= Answer 1 Question 7 , 𝑦= Answer 2 Question 7 , and 𝑓(𝑥,𝑦)= Answer 3 Question 7

Use Lagrange multipliers to find the optimal value of the function 𝑓(𝑥,𝑦)=𝑥2+𝑦2+𝑥𝑦 subject to the constraint 𝑥−2𝑦=7 Find the location of the optimal point, the value of 𝜆 and the value of 𝑓(𝑥,𝑦) at this point. Enter non-integer numerical values as decimals to at least 3 decimal places. Note: you must use a . and not , for a decimal point. The optimal point is located at 𝑥= Answer 1 Question 5 , 𝑦= Answer 2 Question 5 , with 𝜆= Answer 3 Question 5 , and 𝑓(𝑥,𝑦)= Answer 4 Question 5

rite down the Lagrangian function for the following maximization problem.max x + ln y s.t. x2 + y2 ≤ 1, and y ≥ 12

The function 𝑓(𝑥)=−3𝑥4+16𝑥3+24𝑥2−192𝑥+10 has critical points at some of the following 𝑥 -values. In each case identify whether the point is a critical point, and if so find out if it is a maximum, minimum or neither maximum nor minimum. The point 𝑥=−6 is Answer 1 Question 3 The point 𝑥=−2 is Answer 2 Question 3 The point 𝑥=0 is Answer 3 Question 3 The point 𝑥=2 is Answer 4 Question 3 The point 𝑥=4 is Answer 5 Question 3

A consumer spends an amount 𝑚𝑚 to buy 𝑥𝑥 units of one good at the price of 6 per unit and 𝑥𝑥 units of a different good at the price of 10 per unit. Here 𝑚𝑚 is positive and suppose that 8 < 𝑚𝑚 < 40. The consumer’s utility function is 𝑈(𝑥, y) = 𝑥y + y^2+ 2𝑥 + 2y, so that her problem is: Maximize 𝑥y+y^2 +2x+2y subject to 6𝑥+10𝑥y=𝑚 (a) Find the optimal quantities 𝑥∗ and y∗ and the Lagrange multiplier, all of them as functions of 𝑚. (b) Write down the maximum value of the utility function as a function of m. (c) Find the derivative of the above utility function for m=20 (d) What are the solutions for 𝑥∗ and y∗ if (i) 𝑚 ≤ 8? (ii) 𝑚 ≥ 40?