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Homework answers / question archive / Determine the level of production that minimizes the average cost if the cost function is C(x)=10 e0
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
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.
