31. If $p(x)$ is the total value of the production when there are $x$ workers in a plant, then the average productivity of the workforce at the plant is $$A(x)=\frac{p(x)}{x}$$ (a) Find $A^{\prime}(x)$. Why does the company want to hire more workers if $A^{\prime}(x)>0 ?$ (b) Show that $A^{\prime}(x)>0$ if $p^{\prime}(x)$ is greater than the average productivity.

Problem 31E

31. If $p(x)$ is the total value of the production when there are $x$ workers in a plant, then the average productivity of the workforce at the plant is

$$A(x)=\frac{p(x)}{x}$$

(a) Find $A^{\prime}(x)$. Why does the company want to hire more workers if $A^{\prime}(x)>0 ?$

(b) Show that $A^{\prime}(x)>0$ if $p^{\prime}(x)$ is greater than the average productivity.

Calculus, James Stewart 8th edition
2. Derivatives
7. Rates of Change in the Natural and Social Sciences
Problem no. 31E from 180

Step-by-Step Solution

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.

Answer

The value of A'(x) is xp'(x)−p(x)x2.

Explanation

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'v−v'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)x−p(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.

To determine

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

Explanation

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.