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Math 76O-250 Calculus for Business
Self-marking on-line assignment 12

Finding Critical Points for Functions of 2 Variables
Best submitted over the internet by 1:00 PM on Tuesday 21 November 2017

Methods will be discussed in class

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Using a graphing calculator is NOT allowed on this practice assignment.

How to Find Critical Points for f ( x, y )
[1] Differentiate f (x, y), to obtain fx and fy

[2] Use algebra to solve the two equations: [fx = 0] and [fy = 0], finding points: (x, y)

For this assignment, stop here. However, eventually, you would test the above solutions:

[3] Differentiate fx and fy again to obtain
    A = fxx
    B = fxy
    C = fyy and
    D = AC - B2

[4] Use A, B, C, and D to test what kind of critical points you found in [2]

[1] (value 10%) For the function f (x, y) = x2 + y2 - 4x + 12
  the one critical point is ( , )

[2] (value 10%) For the function f (x, y) = ex² - 4x + ey² - 6y
      that is, { [ e to the power of (x2 - 4x)] plus [e to the power of (y2 - 6y)] }
  the one critical point is ( , )

[3] (value 10%) For the function f (x, y) = x2 + y3 + 4x - 12y
  the two critical points are :
  (smaller value of y) : ( , )   , and
  (larger value of y) : ( , )

[4] The function f (x, y) = x3 + 2y3 - 3x + 3y2 - 72y +4     has FOUR critical points.
  [4a] (value 10%) the two critical points with smaller x are :
  (with smaller x and smaller y are : ( , )   , and
  (with smaller x and larger y are : ( , )
  [4b] (value 10%) the two critical points with larger x are :
  (with larger x and smaller y are : ( , )   , and
  (with larger x and larger y are : ( , )

[5] (value 10%) For the function f (x, y) = x2 + xy + y2 - 9x + 12
  the one critical point is ( , )

[6] (value 10%) For the function f (x, y) = 4x3+ 6xy - 3y2 - 36x + 12
  the two critical points are :
  (smaller value of x) : ( , )   , and
  (larger value of x) : ( , )

[7] (value 10%) For the function f (x, y) = x3 - 3xy + y3
  the two critical points are :
  (smaller value of x) : ( , )   , and
  (larger value of x) : ( , )

[8] (value 20%, the toughest problem) For the function f (x, y) = -2x4 + 4xy - y2 + 4x - 2y
  the three critical points are :
  (with smallest value of x) : ( , )   , and
  (with middle value of x) : ( , )   , and
  (with largest value of x) : ( , )

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