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Examples with Detailed SolutionsExample 1
Find the critical point(s) of function f defined by
f(x , y) = x2 + y2
Solution to Example 1:
We first find the first order partial derivatives.
fx(x,y) = 2x
fy(x,y) = 2y
We now solve the following equations fx(x,y) = 0 and fy(x,y) = 0 simultaneously.
fx(x,y) = 2x = 0
fy(x,y) = 2y = 0
The solution to the above system of equations is the ordered pair (0,0).
Below is the graph of f(x , y) = x2 + y2 and it looks that at the critical point (0,0) f has a minimum value.
Example 2
Find the critical point(s) of function f defined by
f(x , y) = x2 - y2
Solution to Example 2:
Find the first order partial derivatives of function f.
fx(x,y) = 2x
fy(x,y) = -2y
Solve the following equations fx(x,y) = 0 and fy(x,y) = 0 simultaneously.
fx(x,y) = 2x = 0
fy(x,y) = - 2y = 0
The solution is the ordered pair (0,0).
The graph of f(x , y) = x2 - y2 is shown below. f is curving down in the y direction and curving up in the x direction. f is stationary at the point (0,0) but there is no extremum (maximum or minimum). (0,0) is called a saddle point because there is neither a relative maximum nor a relative minimum and the surface close to (0,0) looks like a saddle.
Example 3
Find the critical point(s) of function f defined by
f(x , y) = - x2 - y2
Solution to Example 3:
We first find the first order partial derivatives.
fx(x,y) = - 2x
fy(x,y) = - 2y
We now solve the following equations fx(x,y) = 0 and fy(x,y) = 0 simultaneously.
fx(x,y) = - 2x = 0
fy(x,y) = - 2y = 0
The solution to the above system of equations is the ordered pair (0,0).
The graph of f(x , y) = - x2 - y2 is shown below and it has a relative maximum.
Example 4
Find the critical point(s) of defined by
f(x , y) = x3 + 3x2 - 9x + y3 -12y
Solution to Example 4:
The first order partial derivatives are given by
fx(x,y) = 3x2 + 6x - 9
fy(x,y) = 3y2 - 12
We now solve the equations fx(x,y) = 0 and fy(x,y) = 0 simultaneously.
3x2 + 6x - 9 = 0
3y2 - 12 = 0
The solutions, which are the critical points, to the above system of equations are given by
(1,2) , (1,-2) , (-3,2) , (-3,-2)
Exercises
Find, if any, the critical points to the functions below.
1. f(x , y) = 3xy - x3 - y3
2. f(x , y) = e(- x2 - y2 + 2x - 2y - 2)
3. f(x , y) = (1/2)x2 + y3 - 3xy - 4x + 2
4. f(x , y) = x3 + y3 + 2x + 6y
Answers to the Above Exercises
1. (0,0) , (1,1)
2. (1,-1)
3. (16,4) , (1,-1)
4. no critical points.
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