Thread: Maximum within given contraints (I think)

1. Maximum within given contraints

I have been asked a question by an indonesian student. This is the original question.

I ran this through 2 translators and they both gave this

The maximum value of f (x, y) = 4x +12 y on x +2y <8; 3x +2y <12; x> 0; y> 0 is

I am not sure what this means.
I assume it means find the maximum value of 4x+12y given that all the other inequalities are constraints
I have graphed the restricted area and
I have looked at the value of 4x+12y on each of the intersection points And the highest value is 48 at (0,4)
I think this is the answer but I think i should have a much better argument as to why.

Would anyone like to offer any suggestions/help?
Thanks.

2. Re: Maximum within given contraints (I think)

'Consider' your function as a straight line $4x+12y=c.$
This is a straight line with a slope of $-1/3,$ and if $c=0$ it passes through the origin.
You're looking to find the maximum value for $c$ (subject to the constraints), so let $c$ increase and see what happens. How far can you slide the line before it leaves the shaded area ?

3. Re: Maximum within given contraints (I think)

That's a "linear programming" problem since the "object function", 4x+ 12y, as well as the boundary conditions, is linear. As BobP says, graphing 4x+ 12y= c, for any value, c, gives a straight line. Different values of c give different lines all parallel. As you "slide" the line one way or the other, you should see that the highest value for c occurs just as the line is leaving the "feasible" area, inside the boundary lines. And that will occur when the line crosses one of the vertices of the area. To find the maximum (or minimum) value of 4x+ 12y in that region, check its values at the vertices.

4. Re: Maximum within given contraints (I think)

Excellent. Thanks!