Answer:
a)![\overline{C(x)} = 500x^{-1} + 300 - 300\dfrac{\ln({x})}{x}](https://tex.z-dn.net/?f=%5Coverline%7BC%28x%29%7D%20%3D%20500x%5E%7B-1%7D%20%2B%20300%20-%20300%5Cdfrac%7B%5Cln%28%7Bx%7D%29%7D%7Bx%7D)
b)![\overline{C(e^{\frac{8}{3}})} = 279.1549](https://tex.z-dn.net/?f=%5Coverline%7BC%28e%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%29%7D%20%3D%20279.1549)
Step-by-step explanation:
Given the cost function as C(x):
![C(x) = 500 + 300x - 300\ln{x} \quad\quad,x\geq1](https://tex.z-dn.net/?f=C%28x%29%20%3D%20500%20%2B%20300x%20-%20300%5Cln%7Bx%7D%20%5Cquad%5Cquad%2Cx%5Cgeq1)
a) Find the average cost function, ![(\overline{C(x)})](https://tex.z-dn.net/?f=%28%5Coverline%7BC%28x%29%7D%29)
if C is the cost of selling x units, The Average can be denoted by:
![\overline{C(x)} = \dfrac{\text{total cost of selling x units}}{\text{x units}}](https://tex.z-dn.net/?f=%5Coverline%7BC%28x%29%7D%20%3D%20%5Cdfrac%7B%5Ctext%7Btotal%20cost%20of%20selling%20x%20units%7D%7D%7B%5Ctext%7Bx%20units%7D%7D)
![\overline{C(x)} = \dfrac{C(x)}{x}](https://tex.z-dn.net/?f=%5Coverline%7BC%28x%29%7D%20%3D%20%5Cdfrac%7BC%28x%29%7D%7Bx%7D)
![\overline{C(x)} = \dfrac{500 + 300x - 300\ln{(x)}}{x}](https://tex.z-dn.net/?f=%5Coverline%7BC%28x%29%7D%20%3D%20%5Cdfrac%7B500%20%2B%20300x%20-%20300%5Cln%7B%28x%29%7D%7D%7Bx%7D)
![\overline{C(x)} = 500x^{-1} + 300 - 300\dfrac{\ln({x})}{x}](https://tex.z-dn.net/?f=%5Coverline%7BC%28x%29%7D%20%3D%20500x%5E%7B-1%7D%20%2B%20300%20-%20300%5Cdfrac%7B%5Cln%28%7Bx%7D%29%7D%7Bx%7D)
this is the average cost function
b) The minimum average cost:
To find the minimum average cost, we'll have to differentiate the average cost function
. and equate it to zero. (like finding the stationary point of any function)
![\dfrac{d}{dx}(\overline{C(x)}) = \dfrac{d}{dx}\left(500x^{-1} + 300 - 300\dfrac{\ln({x})}{x}\right)](https://tex.z-dn.net/?f=%5Cdfrac%7Bd%7D%7Bdx%7D%28%5Coverline%7BC%28x%29%7D%29%20%3D%20%5Cdfrac%7Bd%7D%7Bdx%7D%5Cleft%28500x%5E%7B-1%7D%20%2B%20300%20-%20300%5Cdfrac%7B%5Cln%28%7Bx%7D%29%7D%7Bx%7D%5Cright%29)
![\overline{C'(x)} = -500x^{-2} + 0 - 300\dfrac{x\frac{1}{x} - \ln{(x)}}{x^2}}](https://tex.z-dn.net/?f=%5Coverline%7BC%27%28x%29%7D%20%3D%20-500x%5E%7B-2%7D%20%2B%200%20-%20300%5Cdfrac%7Bx%5Cfrac%7B1%7D%7Bx%7D%20-%20%5Cln%7B%28x%29%7D%7D%7Bx%5E2%7D%7D)
now just simplify:
![\overline{C'(x)} = -\dfrac{500}{x^2}-300\dfrac{1 - \ln{(x)}}{x^2}}](https://tex.z-dn.net/?f=%5Coverline%7BC%27%28x%29%7D%20%3D%20-%5Cdfrac%7B500%7D%7Bx%5E2%7D-300%5Cdfrac%7B1%20-%20%5Cln%7B%28x%29%7D%7D%7Bx%5E2%7D%7D)
![\overline{C'(x)} = -\dfrac{1}{x^2}(500+300(1 - \ln{(x)}))](https://tex.z-dn.net/?f=%5Coverline%7BC%27%28x%29%7D%20%3D%20-%5Cdfrac%7B1%7D%7Bx%5E2%7D%28500%2B300%281%20-%20%5Cln%7B%28x%29%7D%29%29)
we've found the derivative of C(x), now to find the minimum we'll equate this derivative to zero. ![\overline{C'(x)} = 0](https://tex.z-dn.net/?f=%5Coverline%7BC%27%28x%29%7D%20%3D%200)
![0 = -\dfrac{1}{x^2}(500+300(1 - \ln{(x)}))](https://tex.z-dn.net/?f=0%20%3D%20-%5Cdfrac%7B1%7D%7Bx%5E2%7D%28500%2B300%281%20-%20%5Cln%7B%28x%29%7D%29%29)
and now solve for x
![0 = 500+300(1 - \ln{(x)})](https://tex.z-dn.net/?f=0%20%3D%20500%2B300%281%20-%20%5Cln%7B%28x%29%7D%29)
![-500 = 300(1 - \ln{(x)})](https://tex.z-dn.net/?f=-500%20%3D%20300%281%20-%20%5Cln%7B%28x%29%7D%29)
![1 + \dfrac{5}{3}=\ln{(x)}](https://tex.z-dn.net/?f=1%20%2B%20%5Cdfrac%7B5%7D%7B3%7D%3D%5Cln%7B%28x%29%7D)
![\ln{(x)}=\dfrac{8}{3}](https://tex.z-dn.net/?f=%5Cln%7B%28x%29%7D%3D%5Cdfrac%7B8%7D%7B3%7D)
![x=e^{\frac{8}{3}}\approx 14.392](https://tex.z-dn.net/?f=x%3De%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%5Capprox%2014.392)
at this value of x the average cost is minimum.
![\overline{C(x)} = 500x^{-1} + 300 - 300\dfrac{\ln({x})}{x}](https://tex.z-dn.net/?f=%5Coverline%7BC%28x%29%7D%20%3D%20500x%5E%7B-1%7D%20%2B%20300%20-%20300%5Cdfrac%7B%5Cln%28%7Bx%7D%29%7D%7Bx%7D)
![\overline{C(e^{\frac{8}{3}})} = 500{e^{-\frac{8}{3}}} + 300 - 300\dfrac{\ln({e^{\frac{8}{3}}})}{e^{\frac{8}{3}}}](https://tex.z-dn.net/?f=%5Coverline%7BC%28e%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%29%7D%20%3D%20500%7Be%5E%7B-%5Cfrac%7B8%7D%7B3%7D%7D%7D%20%2B%20300%20-%20300%5Cdfrac%7B%5Cln%28%7Be%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%7D%29%7D%7Be%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%7D)
![\overline{C(e^{\frac{8}{3}})} = 500{e^{-\frac{8}{3}}} + 300 - 300\dfrac{8}{3e^{\frac{8}{3}}}}](https://tex.z-dn.net/?f=%5Coverline%7BC%28e%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%29%7D%20%3D%20500%7Be%5E%7B-%5Cfrac%7B8%7D%7B3%7D%7D%7D%20%2B%20300%20-%20300%5Cdfrac%7B8%7D%7B3e%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%7D%7D)
![\overline{C(e^{\frac{8}{3}})} = 34.7417 + 300 -55.5868](https://tex.z-dn.net/?f=%5Coverline%7BC%28e%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%29%7D%20%3D%2034.7417%20%2B%20300%20-55.5868%20)
![\overline{C(e^{\frac{8}{3}})} = 279.1549](https://tex.z-dn.net/?f=%5Coverline%7BC%28e%5E%7B%5Cfrac%7B8%7D%7B3%7D%7D%29%7D%20%3D%20279.1549)
This is the minimum average cost!