Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Science Advisor

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook