APPLICATIONS OF DERIVATIVES
3. The product of two positive numbers is 16. Find the numbers
(i) if their sum is least.
(ii) if the sum of one and the square of the other is least.
He nositive numbers and such that
Answers
Given:
product of 2 positive numbers = 16
To Find:
The number
(i) if their sum is least.
(ii) if the sum of one and the square of the other is the least.
Solution:
let the number be "x".
we have,
product of 2 numbers = 16
other number = 16/x
sum of two number S = x + 16/x
value of S is minimum for x such that
ds/dx = 0.
⇒ 1 - 16 / x² = 0
⇒ x² = 16
⇒ x = 4 (since, A/Q the number is positive)
⇒ other number = 16/4 = 4
two numbers = (4, 4)
Now,
the sum of one and the square of the other is S2 = x² + 16/x
and S2 has leastvalue when ds2/dx = 0.
⇒ ds2 / dx = 2x - 16/x² = 0
⇒ 2x³ - 16 = 0
⇒ x³ = 8
⇒ x = 2
⇒ other number = 16/x = 16/2 = 8
the two numbers = (2,8)
So, the numbers if their sum is least are {4,4} and the number if the sum of one and the square of the other is the least are {2,8}.