Question

Maximize z=200x+300y subject to x+2y<10 ,x+y<6,x>4 and x>0 ,y>0
​

Answer 1

In theory I would have started looking at the critical points of the function, ie where the tangent plane is horizontal. Trick is, since the function is linear in both x and y the tangent plane is the curve itself, and it's never horizontal. We have to parametrize the border and look in there.

From the last two conditions (
`x>0; y>0`) we are looking only at the first quadrant. Let's start drawing lines.

The various restrains are color coded in the graph (
`x+2y<10` is red,
`x+y<6` is blue,
`x>4` is orange). That leads to the right triangle with a grey border that gets parametrized as shown. Due to the linearity of the original function, the maximum is at one of the extremes (upper or lower depending if the coefficient of the parameter is positive or negative). At this point is just calculating the values, pick up the largest, and see which points correspond to that value of the parameter.

Number crunching done, the maximum is at (4,2)