# 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.

**Explanation:**

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/x^{2}

⇒ 1 - 1/x^{2} = 0

Let's solve for 'x'.

⇒ 1 = 1/x^{2}

⇒ x^{2 }= 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