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Mathematical Applications for the ...

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

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BuyFindarrow_forward

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 )   = ( x +   5 ) 3 dollars, where x represents the number of hundreds of units produced, producing how many units will minimize average cost? Find the minimum average cost.

To determine

To calculate: The number of units and minimum average cost for the function C(x)=x3+15x2+75x+125.

Explanation

Given Information:

The provided cost function is:

C(x)=x3+15x2+75x+125

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 provided function:

C(x)=x3+15x2+75x+125

Now, average cost is given by the formula,

So, substitute x3+15x2+75x+125 for C(x) in the above formula to get,

C¯(x)=C(x)x=x3+15x2+75x+125x=x2+15x+75+125x

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)=x2+15x+75+125xddx(C¯)=ddx(x2+15x+75+125x)

Use the quotient rule of derivatives to get,

C¯(x)=ddx(x2)+ddx(15x)+ddx(75)+ddx(125x)

Now use ddxxn=nxn1 as below,

C¯(x)=2<

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