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Math

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Suppose that y≤3⁢x, 2⁢x≤y and 13⁢x+13⁢y≤1 together with 0≤x, 0≤y.
Determine a value of k so that the function f(x,y)=kx=y has a positive maximum value on the region at two corners.

ANS: k= ?

  • Math-unknown encoding - ,

    We cannot read the question because of unknown encoding.
    Please specify character encoding used (language) or post the character before x and y in the constraints.
    If it is √() you can write sqrt() instead.

    Also, check if the last equation has been posted correctly.
    Is it
    f(x,y)=kx=y
    or
    f(x,y)=kx-y
    ?
    Thank you.

  • Math - ,

    Suppose that y≤3x, 2x≤y and 1/3x+1/3y≤1 together with 0≤x, 0≤y.
    Determine a value of k so that the function f(x,y)=kx+y has a positive maximum value on the region at two corners.

    ANS: k= ?

  • Math - ,

    Use your calculator to plot three graphs, as follows:
    http://img203.imageshack.us/i/1299091442.png/
    This way, you have an idea what to look for.

    The lines are:
    (blue) f1(x)=y=3x
    (red) f2(x)=y=2x
    (green) f3(x)=y=x/(3x-1)

    The region that satisfies all the constraints are between the red and blue lines, and below the green curve.

    To find the value of k you need to find the coordinates of the two intersection points of the (green) curve with the straight lines.


    This you can obtain by solving the following equations:
    f1(x)=f3(x) => 3x=x/(3x-1)....(1)
    f2(x)=f3(x) => 2x=x/(3x-1)....(2)

    These are quadratics and provide two roots each:
    Solution to (1): (x=0 or) x=4/9, y=4/3
    Solution to (2): (x=0 or) x=1/2, y=1

    Therefore f(x,y)=kx+y must pass through (4/9,4/3) and (1/2,1) to satisfy the last requirement. The value of k is the slope of the line passing through these two points using the usual formula:
    Slope, k = (y2-y1)/(x2-x1)

    Can you take it from here?

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