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Finding the maximum and minimum values

  1. Feb 21, 2012 #1
    I am having trouble with getting started with this one:

    Find the maximum and minimum values of f(x,y)=x+2y on the disk x^2+y^2 ≤ 1

    I have started like this:
    fx(x,y) = 1
    fy(x,y) = 2

    and then Im lost...How do I solve it?
     
  2. jcsd
  3. Feb 22, 2012 #2
    well, seeing as f(x,y)=0 is simply a line, you can find the intersection points between the line y=-1/2x, and the boundary of the disk, therefore the circle with radius=1.

    If you consider f(x,y) as a scalar field, then it is a family of lines with incline=-1/2. Then, you just need to find the constant c (f(x,y)=c), for which the function is tangent to the boundary of the circle.
     
  4. Feb 24, 2012 #3

    HallsofIvy

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    Staff Emeritus
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    Since those are never 0, there is no solution inside the circle. You need to look on the circle. There [itex]y= \pm\sqrt{1- x^2}[/itex] so f(x,y) becomes [itex]x+ 2\sqrt{1-x^2}[/itex] or [itex]x- 2\sqrt{1- x^2}[/itex]. Differentiate those and see where the derivative is 0. Don't forget to look specifically at the value of f at the points (-1, 0) and (1, 0), the endpoints of those intervals.
     
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