Minimum and maximum

Question

Use linear programming to find the ordered pairs(x,y) that maximize and minimize the objective equation P=-5y+2x subject to the following constraints x≥0, y≥0, x+3≤15 and 4x+y≤16. List the maximum and minimum value respectively. 

Answer 1

Four inequalities should describe a quadrilateral that contains every possible pair (x,y) in the Domain of the function

P = -5y + 2x.  The insight offered by Linear Programming is that the Maximum and Minimum values of the function P will be at one of the four intersections between the boundary equations.

 

x ≥ 0

y ≥ 0

x ≤ 18

and y ≤ -4x + 16

 

The four points of intersection are (0,0) (18,0) (0,16) and (18, - 56), except that the last point never enters into the picture because y ≤ -4x + 16 crosses y≥ 0 at x = 4, so your Domain is really triangular with the points (0,0) (0,16) and (4,0)

 

P(0,0) = -5(0) + 2(0) = 0

P(0,16) = -5(16) + 2(0) = -80

P(4,0) = -5(0) + 2(4) = 8

 

So, (4,0) is the Maximum and (0,16) is the Minimum.