# Calculus

The product of 2 positive numbers is 48. find the value of the numbers if the sum of one of the numbers and the cube of the other is a minimum.

1. 👍
2. 👎
3. 👁
1. a = first number

b = second number

a * b = 48 Divide both sides by a

b = 48 / a

S = the sum of one of the numbers and the cube of the other number

S = a + b ^ 3

S = a + ( 48 / a ) ^ 3

S = a + 110,592 / a ^ 3

S = a + 110,592 * a ^ - 3

First derivaton :

d S / d a = 1 - 3 * 110,592 * a ^ - 4

d S / d a = 1 - 331,776 / a ^ 4

Second derivation :

d ^ 2 S / d a ^ 2 = - 3 * 331,776 ( - 4 ) * a ^ - 5

d ^ 2 S / d a ^ 2 = 1,327,104 / a ^ 5

A function has minimum or maximum in poit where first derivation = 0

If second derivaton < 0 function has maximum.

If second derivaton > 0 function has minimum.

In this case:

d S / d a = 1 - 331,776 / a ^ 4 = 0

1 = 331,776 / a ^ 4 Multiply both sides by a ^ 4

a ^ 4 = 331,776

a = fourth root of 331,776

a = + OR - 24

For a = - 24

d ^ 2 S / d a ^ 2 = 1,327,104 / a ^ 5 =

1,327,104 / - 7,962,624 = - -0.166667 < 0

function has maxsimum.

For a = 24

d ^ 2 S / d a ^ 2 = 1,327,104 / a ^ 5 =

1,327,104 / -7,962,624 = 0.166667 > 0

function has minimum.

So a = 24

b = 48 / a = 48 / 24 = 2

The mumbers are a = 24 and b = 2

Local minimum = a + b ^ 3 = 24 + 2 ^ 3 = 24 + 8 = 32

1. 👍
2. 👎
2. Thank you!!!

1. 👍
2. 👎

## Similar Questions

1. ### math

Can someone help me with prime and composite numbers? Prime numbers are counting numbers that can be divided evenly bt only two numbers:1 and themselves. A prime number can also be described as a counting number with exactly two

2. ### math

Find two positive numbers such that the sum of the first and twice the second is 100 and their product is a maximum.

3. ### Maths

find two positive real numbers x and y such that their product is 800 and x+2y is as small as possible

4. ### mathematics

Find two positive numbers whose product is 81 and whose sum is a minimum. (If both values are the same number, enter it into both blanks.)

1. ### calculus

Find two positive numbers whose product is 100 and whose sum is a minimum.

2. ### Geometry

Find one counterexample to show that each conjecture is false. The product of two positive numbers is greater than either number.

3. ### calculus

The sum of two positive numbers is 100. Let one of the numbers be x and express the product P of the two numbers as a function of x.

4. ### math

The sum of two positive numbers is 90. Find a function that models their product P in terms of x, one of the numbers.

1. ### algebra

1. The geometric mean between the first two terms in a geometric sequence is 32. If the third term is 4, find the first term. 2. Insert a geometric mean between k and 1/k. 3. If 2 and 3 are two geometric means between m and n,

2. ### Math

The product of two positive number is 56.if their difference is 10,find the numbers.

3. ### maths - HCF

The product of HCF and LCM of two numbers is 119. Find the two numbers if one of the numbers is not 1. Please help me to solve this.

4. ### calculus

find 3 positive real numbers whose sum is 500 and whose product is as large as possible.