   Chapter 10, Problem 14T ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# The cost of producing x units of a product is given by C ( x ) = 100 + 20 x + 0.01 x 2 How many units should be produced to minimize average cost?

To determine

To calculate: The number of units of the product to be produced to minimize average cost where the cost function is C(x)=100+20x+0.01x2 where x is the number of units produced.

Explanation

Given Information:

The provided cost function is C(x)=100+20x+0.01x2 where x is the number of units produced.

Formula used:

To find relative maxima and minima of a function,

1. Set the first derivative of the function to zero, f(x)=0, to find the critical values of the function.

2. Substitute the critical values into f(x) and calculate the critical points.

3. Calculate the sign of the function to the left and right of the critical values.

(a) If f(x)<0 and f(x)>0 on the left and right side respectively of a critical value, then the point is relative minimum.

(b) If f(x)>0 and f(x)<0 on the left and right side respectively of a critical value, then the point is relative maximum.

Calculation:

Consider the provided cost function is C(x)=100+20x+0.01x2 where x is the number of units produced.

Divide the cost function by x, the number of units produced to obtain the average cost function.

C¯(x)=100+20x+0.01x2x=100x+20xx+0.01x2x=100x+20+0.01x

Calculate the first derivative of the function C¯(x)=100x+20+0.01x.

C¯(x)=100x2+0.01

Set the first derivative equal to 0.

100x2+0.01=0

Solve for x.

0.01=100x2x2=1000

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