Results 1 to 4 of 4

Math Help - Extrema function 2 variables

  1. #1
    Newbie
    Joined
    Dec 2012
    From
    be
    Posts
    19

    Extrema function 2 variables

    Hello!

    I need to calculate the stationary points and determine their nature of f(x,y) = x + ysin(x).

    I'm a little confused by setting δf/δx and δf/δy equal to zero and by that finding the stationary points...

    Can someone solve this for me? Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: Extrema function 2 variables

    What do you get when you equate the first partials to zero?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Dec 2012
    From
    be
    Posts
    19

    Re: Extrema function 2 variables

    Quote Originally Posted by MarkFL2 View Post
    What do you get when you equate the first partials to zero?
    Here's what I got:

    f(x,y) = x + ysin(x)

    ∂/∂x = 1 + ycos(x)
    ∂/∂y = sin(x)

    So,

    1 + ycos(x) = 0 -> y = 1 when even and -1 when odd
    sin(x) = 0 -> x = kπ

    -> Critical points (kπ,1) and (kπ, -1)

    Second derivative test for P(kπ,1) :
    ∂/∂x = ysin(x) -> in P: -1sin(π) = 0
    ∂/∂y = 0 -> in P: 0
    ∂/∂x∂y = cos(x) -> in P: cos(kπ) = -1 or 1

    So Δ = (∂/∂x)(∂/∂y) - (∂/∂x∂y) = (0)(0) - (1) = 0 - 1 = -1
    Thus Δ < 0 -> P(kπ,1) is a saddle point.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: Extrema function 2 variables

    I agree with your first partials:

    f_x(x,y)=1+y\cos(x)=0

    f_y(x,y)=\sin(x)=0

    However, from these I get the critical points:

    (2k\pi,-1),\,((2k+1)\pi,1)

    The sign of the y-coordinate is reversed from what you stated, but this doesn't actually affect the conclusion drawn from the second partials test.

    f_{xx}(x,y)=-y\sin(x) (note we need the negative sign as \frac{d}{dx}(\cos(x))=-\sin(x).

    f_{yy}(x,y)=0

    f_{xy}(x,y)=\cos(x)

    And so:

    D(x,y)=-\cos^2(x) thus:

    D(2k\pi,-1)=D((2k+1)\pi,1)=-1

    Hence, none of the critical points are extrema.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. function local extrema etc
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 30th 2010, 03:44 AM
  2. extrema of trig function
    Posted in the Calculus Forum
    Replies: 3
    Last Post: November 5th 2009, 06:45 AM
  3. Local extrema in multiple variables
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 21st 2009, 05:06 AM
  4. Absolute Extrema (two variables)
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 13th 2009, 01:24 PM
  5. Extrema (Absolute) - Two variables
    Posted in the Calculus Forum
    Replies: 5
    Last Post: April 15th 2007, 08:22 PM

Search Tags


/mathhelpforum @mathhelpforum