The manufacturer of GRIPPER tires modeled its return to sale

The manufacturer of GRIPPER tires modeled its return to sales from television advertising expenditures in two regions, as follows: Region 1: S_1 = 50 + 28x_1 - 0.4x_1^2 Region 2: S_2 = 30 + 26x_2 - 1.1x2^2 where S_1 and S_2 are the sales revenue in millions of dollars and x_1 and x_2 are millions of dollars of expenditures for television advertising. (Round your answers to three decimal places.) (a) What advertising expenditures would maximize sales revenue in each district? Region 1 $ 35 million Region 2 $ million

Solution

a)

region1:

S₁=50+28x₁-0.4x₁²

differentiate with respect to x₁

dS₁/ dx₁ =0+28 -0.8x₁

dS₁/ dx₁ =28 -0.8x₁

critical point occurs when dS₁/ dx₁ =0

28 -0.8x₁=0

x₁=28_/0.8

x₁=35

dS₁/ dx₁ =28 -0.8x₁

d²S₁/ dx₁² = -0.8<0

35 million $ advertising expensitures would maximise sales revenue in region 1

region2:

S₂=30+26x₂-1.1x₂²

differentiate with respect to x₂

dS₂/ dx₂ =0+26 -2.2x₂

dS₂/ dx₂ =26 -2.2x₁

critical point occurs when dS₂/ dx₂ =0

26 -2.2x₂=0

x₂=26_/2.2

x₂=11.818

dS₂/ dx₂ =26 -2.2x₁

d²S₂/ dx₂² = -2.2<0

11.818 million $ advertising expensitures would maximise sales revenue in region 2