Determine the level of production that minimizes the average cost if the cost function is C(x)=10 e0

Accounting Dec 07, 2020

Determine the level of production that minimizes the average cost if the cost function is C(x)=10 e0.04xC(x)=10 e0.04x. Round answer to the nearest whole number.

a. 16 units16 units

b. 20 units20 units

c. 25 units25 units

d. 27 units

Expert Solution

First, given the cost function, we can write average cost function: ¯C(x)=C(x)x=10e0.04xxC¯(x)=C(x)x=10e0.04xx, with the help of the first derivative, we determine the critical points:

¯C(x)=10e0.04xx¯C′(x)=10e0.04x⋅0.04x−10e0.04x⋅1x2¯C′(x)=10e0.04x⋅(0.04x−1)x2=00.04x−1=0x=10.04=25unitsC¯(x)=10e0.04xxC¯′(x)=10e0.04x⋅0.04x−10e0.04x⋅1x2C¯′(x)=10e0.04x⋅(0.04x−1)x2=00.04x−1=0x=10.04=25units

We can prove, we have a maximum:

{x<25→¯C′(x)>0x>25→¯C′(x)<0→maximum{x<25→C¯′(x)>0x>25→C¯′(x)<0→maximum

So, the correct answer is c.

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