Question

Suppose that a laboratory test found the combined antibiotic effect of x $2ri73+++sX2i7Uscr0iIUAAAAASUVORK5CYII=$ milligrams of medicine A and y milligrams of medicine B to be given by the function: $LJjlyGBQt2YAAAAASUVORK5CYII=$

1. Find the amounts of the two medicines that maximize the antibiotic effect. You may assume concavity. (5 pts)

Answer 1

Answer:

Given Combined antibiotic effect of x milligrams (05x555) of medicine A and y milligrams (OS S60) of medicine B is represented by the function f(x,y)=xy – 2x² - y2 +110x+60y ...... (1)

Differentiating the function f(x,y) partially with respect to x and y gives DE 1(x,y) =-4x+y+110; D = S(x,y) = x-2y+60 Solving the simultaneous equations f = 0 and 1 = 0 gives 60+220 110+240 +31 > XY 1 280 3507 x = 40; y = 50 Hence the function f(x,y) has one critical point x = 40, y = 50. It remains to ascertain that this critical value leads to maximum effect of the combined medicines.

The second order partial derivatives of the function P(x, y) are given by

  $1585251085629_image.png$

Since the second order partial derivatives are all constant, applying D-test to the above values leads to

D= fx19-[1.] = 8-1

Since D>0 and fa<0, by D-test the function f(x,y) has maximum value for x = 40 and y = 50