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Finding max and min of a function of several variables( 1 more time )

  1. Oct 19, 2009 #1
    1. The problem statement, all variables and given/known data

    Find min/max/saddle point of the function :

    F(x,y) = y^2 - 2*y*cos(x) ; 1 <= x <= 7

    F_x = 2y*sin(x)
    F_xx = 2y*cos(x)
    F_xy = 2cos(x)

    F_y = 2y - 2cos(x)
    F_yy = 2


    Finding its critical points :

    F_x = 2y*sin(x) = 0
    F_y = 2y - 2cos(x) = 0

    Solving for y in F_y

    2y = -2cos(x)
    y = -cos(x)

    inputing this value into F_x

    2( -cos(x) ) * (sin(x) ) = 0;

    realizing trig identities, somehow.

    2( - cos(x) ) * (sin(x) ) = -2sin(x)cos(x) = -2sin(2x)

    so ,

    -2sin(2x) = 0

    That function equals 0 when x = n*pi

    but we have boundaries so ,

    1<= 2n*pi <= 7
    or
    1/2 <= n*pi <= 7/2

    the only solutions is n = 1, so n*pi = pi. ( but the book has critical points as 0,pi,2pi )

    I wont do on because I know I did something wrong. I don't think they accounted for the
    extra 2 in 2n*pi, otherwise the critical points would be 0,pi,2pi.
     
  2. jcsd
  3. Oct 19, 2009 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    so either y= 0 or sin(x)= 0=> [itex]x= n\pi[/itex]

     
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