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.