Calculus
posted by Sunny on .
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.

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 
Thank you!!!