BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 3.8, Problem 31E

(a)

To determine

To find: The value of A'(x) and the reason that why the company wants to hire more workers if A'(x)>0.

Expert Solution

Answer to Problem 31E

The value of A'(x) is xp'(x)p(x)x2.

Explanation of Solution

Given:

The average productivity of the workforce at the plant is shown below with an equation.

A(x)=p(x)x (1)

Calculation:

Calculate the value of A'(x).

Differentiate equation (1) with respect to x.

Apply the quotient rule below.

(uv)'=u'vv'uv2

Substitute p(x) for u and x for v in the above equation.

(p(x)x)=p'(x)x(1)p(x)x2=p'(x)xp(x)x2A'(x)=xp'(x)p(x)x2

If the condition A'(x)>0 is true, then A(x) is increasing. For the condition A'(x)>0 to be true, the total value of production in the plant p(x) should increase. It shows that the average productivity will increase, as the size of the workforce increases. The company will be interested in hiring more workers.

(b)

To determine

To show: The condition A'(x)>0 is true, if the expression p'(x) is greater than the average productivity.

Expert Solution

Explanation of Solution

For the given condition, p'(x) is greater than the average productivity, that is,

p'(x)>A(x) (2)

Substitute p(x)x for A(x) in the equation (2).

p'(x)>p(x)xxp'(x)>p(x)xp'(x)p(x)>0

Multiply by 1x2 on both sides of the above equation.

xp'(x)p(x)x2>0A'(x)>0

Here, the condition A'(x)>0 is true.

Thus, A(x) is increasing that means that the average productivity will increase as the size of the work force increases.

Hence the condition A'(x)>0 is verified, if the value of p'(x) is greater than the average productivity.

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