   Chapter 10.5, Problem 40E ### 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

# Profit An entrepreneur starts new companies and sells them when their growth is maximized. Suppose that the annual profit for a new company is given by P ( x ) = 42 − 1 2 x − 18 x + 1 where P is in thousands of dollars and x is the number of years after the company is formed. If she wants to sell the company before profits begin to decline, after how many years should she sell it?

To determine

To calculate: The number of years that the entrepreneur should wait to sell the company if the annual profit for a new company is given by P(x)=4212x18x+1.

Explanation

Given Information:

The annual profit for a new company is given by,

P(x)=4212x18x+1

Where P is in thousands of dollars and x is the number of years after the company is formed.

Formula used:

To find relative maxima or minima of a function,

1. Find the first derivative of the function.

2. Set the derivative equal to 0 and solve for values of x then satisfy f(x)=0.

3. Evaluate f(x) at a value of x to the left and one to the right of each critical point to develop a sign diagram.

If f(x)>0 to the left and f(x)<0 to the right of the critical value, the critical point is a relative maximum point.

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

Calculation:

As it is provided that the annual profit for a new company is given by,

P(x)=4212x18x+1

Where, P is in thousands of dollars and x is the number of years after the company is formed.

Now, obtain the derivative of this function as,

P(x)=12+18(x+1)2

Set P(x)=0 to obtain the critical points as,

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