Minimum average cost
the cost of producing x units of a product is modeled by
c = 100 + 25x - 120 ln x, x ≥ 1.
(a) find the average cost function ć.
(b) find the minimum average cost analytically. use a graphing utility to confirm your result.

a)

b)

Step-by-step explanation:

Given that

C = 100 + 25 x - 120 ln x   ,x ≥ 1.

The average cost function given as

Therefore

To find average minimum cost

100 + 120 (1-lnx) = 0

ln x =1.833

x=6.25

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.