Question

In order to treat a certain bacterial infection, a combination of two drugs is being tested. Studies have shown that the duration of the infection in laboratory tests can be modeled by D(x, y) = x2 + y2 - 12x - 30y + 2xy + 140 where x is the dosage of the first drug and y is the dosage of the second drug (both in hundreds of milligrams). Find the amount of each drug necessary to minimize the duration of the infection. first drug hundred mg second drug hundred mg Need Help? Read Watch Master it Talk to a Tutor

Answer 1

Given; Deasy) = x² +442 -12x-30y + 2xy +140 Dx=2x-12 +24 { doing partial differentiation with respect tox ey } Dy = oy -30+2x - 8 & Dxx=2 Dyy Dry=2 Now To determine critical numbers Put Dx=0 & Dy zo ie. 2x-12+2y=o 284-30+2x20 eq 0 -eq gives; 2x-12 +2y - oy +30 - 2x = 0 -by= 10 =) y = 3 from eq 0, 2x - 12 +6=0 2x=6 = x= 3 = critical points to yo) =(3, 3)

Now Determinant, 2 -Day D= Dax Dyy 2X8 22 D = 12 Sos el congo = 12 (20.yo) and Dxa 2 7o (20. Yo ) =) D has relative minimum at (3,3)=(2, Yo). Therefore; The amount of each drug. necessary to minimize the duration of the infection is; hundred mg. first drug, x = 3 hundred mg second drug, y= 3