Find a positive number such that the sum of the number and its reciprocal is as small as possible.
Optimization is a mathematical technique to find the minimum or maximum values of a function.
Answer: The required positive number is 1.
Let's find the sum of a positive number and its reciprocal.
Let one positive number be 'x' and its reciprocal be 1/x where x ≠ 0.
We need to find the sum of numbers.
⇒ f (x) = x + 1/ x
To find the minimum value of f (x), we will optimize the function and differentiate it with respect to 'x'. and d/dx = 0
⇒ dS/ dx = 1 - 1/x2
⇒ 1 - 1/x2 = 0
Let's solve for 'x'.
⇒ 1 = 1/x2
⇒ x2 = 1
⇒ x = ± 1
Since we cannot have a negative value for x and x > 0 Therefore, x = 1.
By substituting the value x in f (x), we get
f (1) = 1 + 1/1 = 2