Calculus in economics

Question

Problem Set question:

 

The cost, in dollars, of producing x units of a certain item is given by

C(x)= 5x−4 sqrt(x-4).

 

Find the production level that minimizes the average cost per unit.

 

 

The number of units that minimizes the average cost is ____________.

Answer 1

C(x) = 5x - 4*sqrt(x-4)

Note: x >= 4

Find the derivative and set it = 0

this will identify extrema, where slope = 0

C'(x) = 5 - (4)(1/2)(x-4)^(-1/2)

C'(x) = 5 - 2/sqrt(x-4)

0 = 5 - 2/sqrt(x-4)

2/sqrt(x-4) = 5

sqrt(x-4)/2 = 1/5

sqrt(x-4) = 2/5

x-4 = 4/25

x = 4 + 4/25 = 104/25

Find second derivative

to determine concavity.

This lets us know if it is a max or min.

C''(x) = (-2)(-1/2)(x-4)^(-3/2)

C''(x) = (x-4)^(-3/2)

Second derivative is positive for all x >=4

So, the extrema is a minimum

To find the minimum cost,

Evaluate the original cost equation

for x = 104/25

C(x) = 5x - 4*sqrt(x-4)

C(104/25) = 5(104/25) - 4*sqrt(104/25 - 4)

C(104/25) = 104/5 - 4*sqrt(4/25)

C(104/25) = 104/5 - 4(2/5)

C(104/25) = 104/5 - 8/25

C(104/25) = 96/5

C(104/25) ~ 19.2