Labware - MA35 Multivariable Calculus - Two Variable Calculus



Critical Points (Page: 1 | 2 )


A critical point of a function of two variables f(x,y) is a point such that the tangent plane is horizontal.



  • 1. Note the clockwise order of the colors around the saddle points in the graph; red, purple, blue, white. Now enter f(x,y) = x2 + y2. Is the sequence red, purple, blue, white in clockwise or counter-clockwise order around the minimum in this graph? What about for the maximum in f(x,y) = -x2 - y2?
  • 2. What does the orientation of the colors tell you about the type of critical point?
  • 3. Try entering f(x,y) = -x4 + x2 - y2, which has three critical points visible. Based on your answer to the last exercise, what type of critical points are each of these three?
  • 4. Why do critical points only occur at corners which border all four colors?
  • 5. Analyze the function f(x,y) = (y-x2)(y-1). What can we say about the 0 level sets of f(x,y), fx(x,y), and fy(x,y)?
  • 6. Analyze Crater Lake shifted by an earthquake, with function f(x,y) = -(x2 + y2)2 + 2(x2 + y2) + mx for various values of m. For which m will the lake no longer hold water? Describe the critical levels, i.e. the level sets that contain critical points.