M ASSACHUSETTS I NSTITUTE OF T ECHNOLOGY Interphase Calculus III Worksheet Instructor: Samuel S. Watson 24 July 2015 Topics: Lagrange multipliers, critical points of functions with two variables 1. (a) Define f ( x, y) = x2 + y2 , and suppose that g is a function such that the graph of g( x, y) = c is the heart-shaped curve below. Identify the point Pmax on the curve g( x, y) = c for which f is maximal. Identify the point Pmin on the curve where f is minimal. (b) Draw the level curve of f which passes through Pmax and the level curve of f which passes through Pmin . (c) Describe the relationship between the two level surfaces which intersect at Pmax or at Pmin in your picture above. Use that relationship to write down a system of three equations satisfied by the coordinates of Pmax and Pmin . This is called the method of Lagrange multipliers. 2. Use the method of Lagrange multipliers to find the maximum and minimum values of the function f ( x, y) = x2 + 2y2 on the circle x2 + y2 = 1. 3. Use the method of Lagrange multipliers to find the maximum volume of a rectangular box without a lid which can be made from 12 square meters of cardboard. 4. Apply the second derivative test to try to classify the critical point at x = 0 for each of the functions f ( x ) = x2 , g( x ) = − x2 , h( x ) = x3 , and k ( x ) = x4 . Your answer for each should be local minimum, local maximum, or inconclusive. 5. A function f ( x, y) has a local minimum at ( x0 , y0 ) if there exists a disk centered at ( x0 , y0 ) throughout which the function’s values are all greater than or equal to f ( x0 , y0 ). In this problem, we figure out how to tell whether a twice-differentiable function has a local minimum or local maximum at a point. (a) Suppose that f has a local minimum at a point ( x0 , y0 ), and suppose ` is any line in the xyplane passing through ( x0 , y0 ). What can we conclude about the one-variable function obtained by restricting f to `? (b) We want to check the restriction of f to every line ` has a local minimum at ( x0 , y0 ). Let u = (h, k ) be a unit vector, and let g(t) = f (( x0 , y0 ) + tu) be the function obtained by restricting f to the line through ( x0 , y0 ) in the direction of u. Use the chain rule to find the second derivative of g at t = 0. (c) Multiply and divide by h2 , then substitute m = k/h to get a second-degree polynomial in m. (d) Use your knowledge of parabolas to find a condition to check whether this expression is always positive, always negative, or sometimes positive and sometimes negative, as m varies. 6. Apply the second derivative test to the functions x2 + y2 , − x2 − y2 , and xy to classify the critical point at the origin. The graph of the function xy is shown below along with several traces. 7. Classify the critical points of f ( x, y) = x4 + y4 − 4xy + 1. 8. (Challenge problem) The Hardy-Weinberg law states that the proportion of individuals in a population who carry two different blood-type-related alleles is P = 2pq + 2pr + 2rq, where p, q, and r are the proportions of the three different alleles in the population. Show that P is at most 2/3, given that p + q + r = 1.