Results 1 to 3 of 3

Math Help - Stationary Points

  1. #1
    Newbie
    Joined
    Nov 2007
    Posts
    20

    Stationary Points

    Happy new year guys! Don't ask me why I'm doing maths on New Years Eve

    Question:
    Find the stationary points and examine the following function for relative extrema and saddle points:
     f(x,y) = 4x^2 e^y - 2x^4 - e^{4y}

     f_x = 8xe^y - 8x^3 = 0 which implies x=0 or  x=e^{\frac{y}{2}}
     f_y = 4x^2e^y - 4e^{4y} = 0

    substituting x=0 gives you end up getting  e^{4y} = 0 which is not defined. substituting   x=e^{\frac{y}{2}} gives you  4e^{2y}(1-e^{2y}) = 0 which implies y=0.

    So (1,0) is a stationary point - I am able to find the nature of this point, so don't worry about helping me with that.

    From the  f_y = 0 equation, you get  4e^y (x^2 - e^{3y} ) = 0 , so  x = e^{\frac{3y}{2}} , substituting into the  f_x equation, you get  8e^{\frac{5y}{2}} - 8e^{\frac{15}{2}}=0
     y= \frac{2}{5} = 3

    So  ( e^{\frac{9}{2}}, 3) is another stationary point. Again I'm able to find the nature of this point.

    These seem right because I've substituted them into the  f_x and  f_y equations and they work.

    My concern is there must be more, because looking at the graph, there seem to be at least 2 maximum points and a minimum. Have I gone wrong somewhere or missed something? Thanks for taking the time to look at this.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Square roots can be negative!

    From the first partial derivative, x=0 or ħe^{y/2}. From the second partial derivative, x=ħe^{3y/2}.

    (And don't ask what I'm doing for New Year's Eve!)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2007
    Posts
    20
    Quote Originally Posted by Opalg View Post
    Square roots can be negative!

    From the first partial derivative, x=0 or ħe^{y/2}. From the second partial derivative, x=ħe^{3y/2}.

    (And don't ask what I'm doing for New Year's Eve!)
    Thank you!!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: August 24th 2011, 12:35 PM
  2. stationary points
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 16th 2009, 06:00 AM
  3. Stationary Points
    Posted in the Calculus Forum
    Replies: 4
    Last Post: December 3rd 2008, 01:18 PM
  4. Stationary Points, Local points of Extrema
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 19th 2008, 02:44 PM
  5. Replies: 3
    Last Post: May 5th 2006, 10:22 AM

Search Tags


/mathhelpforum @mathhelpforum