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Finding Critical Points Of A Function

  • Thread starter Lancelot59
  • Start date
  • #1
634
1
I'm trying to find the minimia and maxima of the following function without using LaGrange multipliers:

[tex]f(x,y)=sin(x)+sin(y)+sin(x+y)[/tex]
where:
[tex]0\leq x \leq 2\Pi[/tex]
[tex]0\leq y \leq 2\Pi[/tex]

Partial derivatives:
[tex]f_{x}=cos(y)+cos(x+y)[/tex]
[tex]f_{y}=cos(x)+cos(x+y)[/tex]

[tex]f_{xx}=-sin(x+y)[/tex]
[tex]f_{xy}=-sin(y)-sin(x+y)[/tex]
[tex]f_{yy}=-sin(x+y)[/tex]

Now I have no clue how to get all the critical points. I simplified it using fx=0, fy=0, to get fx=fy. Equating and simplifying I got cos(x)=cos(y), x=y.

Now this is where I get lost. How do you pick points to try and solve the equations? I could say that cos(x)=cos(y), then cos(x)-cos(y)=0. So you could pick x or y to be pi/2 or 3pi/2.

However the solution manual to my textbook does not use any of these points...so how does this work?
 

Answers and Replies

  • #2
33,636
5,296
I'm trying to find the minimia and maxima of the following function without using LaGrange multipliers:

[tex]f(x,y)=sin(x)+sin(y)+sin(x+y)[/tex]
where:
[tex]0\leq x \leq 2\Pi[/tex]
[tex]0\leq y \leq 2\Pi[/tex]

Partial derivatives:
[tex]f_{x}=cos(y)+cos(x+y)[/tex]
[tex]f_{y}=cos(x)+cos(x+y)[/tex]
You got off to a bad start. The two first partials are incorrect.
fx = cos(x) + cos(x + y)
fy = cos(y) + cos(x + y)

Because of this error, there are errors in your second partials as well.
[tex]f_{xx}=-sin(x+y)[/tex]
[tex]f_{xy}=-sin(y)-sin(x+y)[/tex]
[tex]f_{yy}=-sin(x+y)[/tex]

Now I have no clue how to get all the critical points. I simplified it using fx=0, fy=0, to get fx=fy. Equating and simplifying I got cos(x)=cos(y), x=y.

Now this is where I get lost. How do you pick points to try and solve the equations? I could say that cos(x)=cos(y), then cos(x)-cos(y)=0. So you could pick x or y to be pi/2 or 3pi/2.

However the solution manual to my textbook does not use any of these points...so how does this work?
 
  • #3
634
1
You got off to a bad start. The two first partials are incorrect.
fx = cos(x) + cos(x + y)
fy = cos(y) + cos(x + y)

Because of this error, there are errors in your second partials as well.
WolframAlpha agrees with me.
Partial X
Partial Y
 
  • #4
Mute
Homework Helper
1,388
10
WolframAlpha agrees with me.
Partial X
Partial Y
Take a careful look at what you wrote for the first derivatives in your first post, what Mark44 wrote, and what wolframalpha is giving you. I assure you wolframalpha agrees with Mark44 (and that the error affects your second derivatives).
 
  • #5
634
1
DAH! That would do it...

So if fx=fy

cos(x)+cos(x+y)=cos(y)+cos(x+y)
cos(x)=cos(y)
x=y

Now what?
 
  • #6
Mute
Homework Helper
1,388
10
DAH! That would do it...

So if fx=fy

cos(x)+cos(x+y)=cos(y)+cos(x+y)
cos(x)=cos(y)
x=y

Now what?
Plug y=x back into f_x=0 to figure out what x (and hence y) must be. To determine if the point is a min or a max you need to recompute the partial derivatives, as the second derivatives in the first post are incorrect because you initially mixed up f_x and f_y. Then, calculate the value of the Hessian matrix to see if it's a min, max or saddle.
 
  • #7
634
1
I got it. Thanks!
 

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