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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 3:00 PM on Tuesday 20 November 2018

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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