1. Let C(x) denote the cost of producing x units of a certain product. Assume th

Question

1. Let C(x) denote the cost of producing x units of a certain product. Assume that C'' (x) > 0 for all x > 0. For x > 0, let A(x) = C(x)/x .

(a) Explain that A(x) denotes the average cost to produce each unit if x units are produced.

(b) Show that the average cost is minimized at a production level x = b when A(b) = C'(b). Be sure to justify that the production level you found gives a minimum value of the average cost, and not a maximum value.

(c) Show that the line through (0, 0) and (b, C(b)) is tangent to the graph of C(x) at x = b.

2. Find the coordinates of a point on the graph of y = sqrt(x ? 1) that is closest to the point (3, 0).

Explanation / Answer

1) solution:

c(x) = cost of producing x units of product

given A(X) = C(X)/X

(a) average cost = cost of producing x units of product / number of units of product

                         = C(X) / X

So, A(X) is average cost to produce each cost.

(b) A(X) = C(X)/X

for minimization differentiate it and equate it to zero.

A'(X) = [ X C'(X) - C(X) ] / X2

SO, A'(X) = 0

So, X C'(X) - C(X) =0

So, C(X) = X C'(X)

AT a production level x=b,

C(b) = b c'(b)    where A(b) = C(b) / b

so, A(b) = C'(b)

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