   Chapter 10.3, Problem 16E ### 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

# If the total cost function for a product is C ( x )   =   300   +   10 x +   0.03 x 2 dollars, producing how many units, x, will result in a minimum average cost per unit? Find the minimum average cost.

To determine

To calculate: The minimum average cost and number of units that is x for the cost function C(x)=300+10x+0.03x2.

Explanation

Given Information:

The provided cost function is:

C(x)=300+10x+0.03x2

Formula used:

If f(x) and g(x) are two differentiable functions then by the property of derivative:

ddx(f(x)+g(x))=ddxf(x)+ddxg(x)

And

ddxxn=nxn1

Where n is a constant and x is the variable.

And the average cost is defined as:

C¯(x)=C(x)x

Where C(x) is the cost function.

Calculation:

Consider the cost function C(x)=300+10x+0.03x2.

Average cost is given as:

C¯(x)=C(x)x=300+10x+0.03x2x=300x+10+0.03x

The absolute maxima and absolute minima will occur only at the critical points. To calculate the critical points of the average cost function find the first derivative of the function:

C¯(x)=300x+10+0.03xddx(C¯)=ddx(300x+10+0.03x)

Use ddx(f(x)+g(x))=ddxf(x)+ddxg(x)

R¯(x)=ddx(300x1)+ddx(10)+ddx(0

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