Question

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= ?

Answer 1

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.

Answer 2

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= ?

Answer 3

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?