Math 21a, Fall 2008
Problems A, Math 21a, Multivariable Calculus
Monkey saddle
Course head: Oliver Knill
Office: SciCtr 434
Problem A:

The graph of the function 6 x y2 - 2 x3 - 3 y4 is called a Monkey saddle. We want to find the global maxima and minima of the function f(x,y) on the unit disc x2+y2 less or equal to 1. A point (a,b) is a global maximum on the unit disc G, if f(x,y) less or equal to f(a,b) for all (x,y) on G. To do so:

a) (3 points) Find the critical points of f in the interior of the disc. Can you apply the second derivative test?

b) (4 points) Find the critical points on the boundary of the disc using the Lagrange multiplier method.

c) (3 points) Where are the global maxima and minima of the function on the disc? To find the global maxima and minima, you have to look at all critical points in the interior as well as the extrema on the boundary. Just throw them all in the pot and look which is the biggest.

Optional and not required: make a plot of the surface using Mathematica. Here is an example on how to plot a graph of a function of two variables:
Plot3D[x^2+y^2,{x,-1,1},{y,-1,1}]


Questions and comments to knill@math.harvard.edu
Math21b | Math 21a | Fall 2008 | Department of Mathematics | Faculty of Art and Sciences | Harvard University