Suppose the fraction of Mazda car owners who chose another new Mazda can be mode

Question

Suppose the fraction of Mazda car owners who chose another new Mazda can be modeled by the following function:

M(c, f) = 10 8f 4f2 + 40c + 20fc 40c2

where c is the fraction of Chrysler car owners who remained loyal to Chrysler and f is the fraction of Ford car owners remaining loyal to Ford. Locate and classify all the critical points.

(c,f)=(____)

Interpret your answer. (Enter your answers as fractions.)

At most, _____of all Mazda owners would choose another new Mazda, and this  ---Select--- (highest/ lowest)  loyalty occurs when ____of Chrysler and ____of Ford owners remain loyal to their brands.

Explanation / Answer

M(c, f) = 10 8f 4f2 + 40c + 20fc 40c2

Mc = 0 0 0 + 40 + 20f 80c

Mc = 40 + 20f 80c

Mc = 0

==>4c-f=2

==>f=4c-2------>(1)

Mf = 0 8 8f + 0 + 20c 0

Mf = 8 8f + 20c

Mf =0

==>20c-8f=8

==>5c-2f=2------>(2)

put(1) in(2)

5c-2(4c-2)=2

-3c+4=2

c=2/3

f=4(2/3)-2 =2/3

critical point (2/3,2/3)

Mcc=-80,Mff=-8,Mcf=20

D=MccMff -Mcf2

D=640-202

D=240>0,Mcc<0

M(2/3,2/3)=10 8(2/3) 4(2/3)2 + 40(2/3) + 20(2/3)(2/3) 40(2/3)2

At most, 2/3 of all Mazda owners would choose another new Mazda, and this highest loyalty occurs when 2/3 of Chrysler and 2/3 of Ford owners remain loyal to their brands.

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