Problem 5

The sum of two positive numbers is 2. Find the smallest value possible for the sum of the cube of one number and the square of the other.

Solution 5
Let x and y = the numbers

x + y = 2            Equation (1)
1 + y’ = 0
y’ = –1

z = x³ + y²            Equation (2)
dz/dx = 3x² + 2y y’ = 0
3x² + 2y(–1) = 0
y = (3/2) x²

From Equation (1)
x + (3/2) x² = 2
2x + 3x² = 4
3x² + 2x – 4 = 0
x = 0.8685 & –1.5352
use x = 0.8685

y = (3/2)(0.8685)²
y = 1.1315

z = 0.8685³ + 1.1315²
z = 1.9354 answer

Problem 6

Find two numbers whose sum is a, if the product of one to the square of the other is to be a minimum.

Solution:
Let x and y = the numbers

x + y = a
x = a – y

z = xy²
z = (a – y) y²
z = ay² – y³
dz/dy = 2ay – 3y² = 0
y = 2/3 a

x = a – 2/3 a
x = 1/3 a

The numbers are 1/3 a, and 2/3 a. answer

Problem 7

Find two numbers whose sum is a, if the product of one by the cube of the other is to be a maximum.

Solution:
Let x and y the numbers

x + y = a
x = a – y

z = xy³
z = (a – y) y³
z = ay³ – y⁴
dz/dy = 3ay² – 4y³ = 0
y² (3a – 4y) = 0
y = 0 (absurd) and 3/4 a (use)

x = a - 3/4 a
x = 1/4 a

The numbers are 1/4 a and 3/4 a. answer

Problem 8

Find two numbers whose sum is a, if the product of the square of one by the cube of the other is to be a maximum.

Solution:
Let x and y the numbers

x + y = a
1 + y’ = 0
y = –1

z = x² y³
dz/dx = x2 (3y² y’) + 2xy³ = 0
3x(–1) + 2y = 0
x = 2/3 y

2/3 y + y = a
5/3 y = a
y = 3/5 a

x = 2/3 (3/5 a)
x = 2/5 a

The numbers are 2/5 a and 3/5 a. answer