Math 21a, Fall 2008

Problems A, Math 21a, Multivariable Calculus

Monkey saddle

Course head: Oliver Knill

Office: SciCtr 434

Email: knill@math.harvard.edu

Problem A: The graph of the function 6 x y ^{2} - 2 x^{3} - 3 y^{4}
is called a Monkey saddle.
We want to find the global maxima and minima of the function f(x,y) on the unit disc
x^{2}+y^{2} 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