Calculus 3 (Math-UA-123) Fall 2016 Homework 6 Due: Thursday, October 27 at the start of class Please give complete, well-written solutions to the following exercises. 1. Consider the function f (x, y) = kx2 + y 2 4xy, where k is some fixed constant. (a) Show that for any value of k, (0, 0) is a critical point of f . (b) Determine the values of k (if any) for which (0, 0) is i. a saddle point, ii. a local maximum, iii. a local minimum. 2. We want to find the absolute maximum value of f (x, y) = x3 3x y 2 + 12 on the domain D = {(x, y) | x2 + y 2 9, y 0}, a half disk. (a) First, find the critical points of f , and determine if any of the critical points is a local maximum in the domain D. (b) Then, find the (absolute) maximum value of f along the boundary of D: the bottom edge of D and along the top arc of D. (Note: some of the values might be a bit messy. Give solutions to two decimal places.) (c) Using parts (a) and (b), find the absolute maximum value of f on D. 3. The figure below shows contours of f , a function of x and y, and the line that satisfies a constraint g(x, y) = c. (a) Does f have a maximum value subject to the constraint g(x, y) = c for x > 0, y > 0? If so, approximate where it is and what its value is. Show all work/justification. (b) Does f have a minimum value subject to the constraint g(x, y) = c for x > 0, y > 0? If so, approximate where it is and what its value is. Show all work/justification. 1 Calculus 3 (Math-UA-123) Fall 2016 4. A firm has three factories each producing the same item. Let x, y, z denote the respective output quantities that the three factories produce in order to fulfill an order of 2000 units in the total, namely x + y + z = 2000. The cost of producing x units of the item in the first factory is C1 (x) = 200 + 1 2 x, 100 the cost of producing y units of the item and z units of the item in the second and third factories are, respectively: C2 (y) = 200 + y + 1 3 y , 300 C3 (z) = 200 + 10z. Use the method of substitution to find the critical point(s) of the total cost function subject to the constraint that the total number of units produced must be equal to 2000. 5. Consider the problem of minimizing the function f (x, y) = x on the curve y 2 +x4 x3 = 0 (a type of curve known as a \"piriform\"). (a) Try using Lagrange multipliers to solve the problem. (b) Show that the minimum value is f (0, 0) even though the Lagrange condition f (0, 0) = g(0, 0) is not satisfied for any value . (c) Briefly explain (1-2 sentences) why Lagrange multipliers fail to find the minimum value in this case. (Hint for parts (b) and (c): try to plot the curve using Wolfram Alpha, and locate where the point (0, 0) is on this curve.) Challenge Problem: Method of Least Squares (NOT to be graded nor turned in) Consider the points (0, 1), (1, 0), (2, 2). We would like to try and find a line of the form f (x) = mx + b which \"fits\" the data as well as possible. One method to do this is the following: for each of the three points (x, y) above compute the value d(x, y) = y (mx + b). For example: d(0, 1) = 1 (m 0 + b) = 1 b. Note that |d| measures the difference between the actual value of y and the one suggested by the line f (x) = mx + b. 2 Calculus 3 (Math-UA-123) Fall 2016 After we compute the values of each d for the three points we would like to solve the following: Find m and b which minimizes |d(0, 1)| + |d(1, 0)| + |d(2, 2)|. Since absolute values are not differentiable at 0, we minimize the sum of the squares of the difference instead: Find m and b which minimizes (d(0, 1))2 + (d(1, 0))2 + (d(2, 2))2 . The solution for m and b above will give us a line f (x) = mx + b which we call the least squares approximation for the data. (a) Solve the minimization problem above (boxed). Plot the three data points and the line that you compute in your solution. (b) Replace the points (0, 1), (1, 0), (2, 2) in part (a) with three general data points (x1 , y1 ), (x2 , y2 ), (x3 , y3 ) and define d1 = d(x1 , y1 ) etc. State the minimization problem you would need to solve to find the least squares line. What system of equations do you derive by taking partial derivatives with respect to m and b? (c) Generalize (b) to n arbitrary data points: (x1 , y1 ), (x2 , y2 ), . . . , (xn , yn ). 3