• Support PF! Buy your school textbooks, materials and every day products Here!

Finding the critical point and its nature. With solid attempt

  • Thread starter tamintl
  • Start date
  • #1
74
0
Finding the critical point and its nature. With solid attempt!!

Homework Statement



Find all critical points of the function

f(x, y) = xy2 - 2xy - 2x2 - 3x +7

and determine their nature.

Homework Equations



none

The Attempt at a Solution



I know that to find the critical points you must set fx = 0 and fy=0

Doing this I get:


fx = -4x + y2 - 2y - 3 = 0

and

fy = 2xy - 2x = 0

Thus, fx = fy

-4x + y2 -2y - 3 = 0

-4x + (y+1)(y-3) = 0

I don't really know how to proceed? Any help would be great.
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618


Factor the fy equation first. What conclusions can you draw from that?
 
  • #3
74
0


Factor the fy equation first. What conclusions can you draw from that?
fy = 2xy - 2x = 2x(y-1)

Thus, x=0 and y=1

SO, critical point is (0,1)...

So is that the only critical point or do I sub this back into fx to get another?

Thanks Dick!
 
  • #4
Dick
Science Advisor
Homework Helper
26,258
618


fy = 2xy - 2x = 2x(y-1)

Thus, x=0 and y=1

SO, critical point is (0,1)...

So is that the only critical point or do I sub this back into fx to get another?

Thanks Dick!
Careful!! 2x(y-1)=0 if x=0 OR y=1. Not necessarily both. Put those two possibilities back into fx and see what happens.
 
  • #5
74
0


Careful!! 2x(y-1)=0 if x=0 OR y=1. Not necessarily both. Put those two possibilities back into fx and see what happens.
Silly me!

Okay subbing back into fx gives me: (0,-1), (0,3), (-1,1)

Assuming these are correct do I now find fxx, fyy, and fxy
to determine the nature (using theorem) ??

On a side note how do you find fxy? I'm confused about that particular instance.

Thanks Dick! I really appreciate your help
 
  • #6
Dick
Science Advisor
Homework Helper
26,258
618


Silly me!

Okay subbing back into fx gives me: (0,-1), (0,3), (-1,1)

Assuming these are correct do I now find fxx, fyy, and fxy
to determine the nature (using theorem) ??

On a side note how do you find fxy? I'm confused about that particular instance.

Thanks Dick! I really appreciate your help
Sure, now use the second derivative test on those three critical points. fxy isn't hard to to find, just take your fx and differentiate with respect to y, OR take fy and differentiate with respect to x. You'll get the same thing.
 
  • #7
74
0


Sure, now use the second derivative test on those three critical points. fxy isn't hard to to find, just take your fx and differentiate with respect to y, OR take fy and differentiate with respect to x. You'll get the same thing.
Got it thanks!

Now I have anther question:

Find all critical points of the function: f(x, y) = y sin x + cos x and determine their nature.

So we have fx = ycosx - sinx and fy = sinx

hence, for

fx we have y = sinx/cosx = tanx

and

fy we have sinx = 0

Therefore we have x=0... when we sub this into y=tanx we get y=0

Hence critical point is (0,0)

Would this be reasonable?

Regards
 
  • #8
Dick
Science Advisor
Homework Helper
26,258
618


Got it thanks!

Now I have anther question:

Find all critical points of the function: f(x, y) = y sin x + cos x and determine their nature.

So we have fx = ycosx - sinx and fy = sinx

hence, for

fx we have y = sinx/cosx = tanx

and

fy we have sinx = 0

Therefore we have x=0... when we sub this into y=tanx we get y=0

Hence critical point is (0,0)

Would this be reasonable?

Regards
x=0 isn't the only solution to sin(x)=0. What are the others?
 
  • #9
74
0


∏, 2∏

Now what :/

edit: x=∏(n) for all n ε Z
 
  • #10
Dick
Science Advisor
Homework Helper
26,258
618


∏, 2∏

Now what :/

edit: x=∏(n) for all n ε Z
Put x=pi*n into fx. What does that tell you about y?
 
  • #11
74
0


Put x=pi*n into fx. What does that tell you about y?
As you put it into fx, sinx = 0... therefore you have fx = -y for odd n and fx = y for even n..

Hence y=0 again for x=n*∏

?
 
  • #12
Dick
Science Advisor
Homework Helper
26,258
618


As you put it into fx, sinx = 0... therefore you have fx = -y for odd n and fx = y for even n..

Hence y=0 again for x=n*∏

?
Fine. So your critical points are (-pi,0), (0,0), (pi,0), (2pi,0)... right? (n*pi,0) where n is any integer.
 
Last edited:

Related Threads on Finding the critical point and its nature. With solid attempt

Replies
8
Views
2K
Replies
5
Views
3K
Replies
5
Views
2K
Replies
1
Views
943
  • Last Post
Replies
3
Views
671
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
7
Views
1K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
3
Views
5K
Top