   Chapter 3.R, Problem 38E

Chapter
Section
Textbook Problem

# Find two positive integers such that the sum of the first number and four times the second number is 1000 and the product of the numbers is as large as possible.

To determine

To find:

Two positive integers such that the sum of the first number and four times the second number is 1000 and product of the numbers is as large as possible.

Explanation

1) Concept:

We apply the second derivative test and critical numbers to find the two positive integers

Second derivative test:

Suppose f'' is continuous near c

If f'c=0 and f''c>0 then f  has local minimum at c

If f'c=0 and f''c<0 then f  has local maximum at c

Critical number:

A critical number of a function f   is a number c in the domain of f  such that either  f'c=0 or f'c does not exist

2) Given:

Conditions on integers:

Sum of the first number and four times the second number is 1000 and product of the numbers is as large as possible.

3) Calculation:

Let x and y be two integers, then by the given condition

x+4y=1000 …………….. (1)

From the above equation, x=1000-4y

Also, from another condition, the product of integers is as large as possible, so the equation becomes

P=xy  ……………(2)

Substitute, x=1000-4y in the above equation.

Therefore

P(y)=1000-4yy  where y=0 y 250

Simplify the right side of the equation,

P(y)=1000y-4y2

Now maximize the function P(y)

Differentiate P(y) with respect to y

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