Overview
Given the problem, that if the product of two positive numbers is 3600, what is the minimum value of their sum. Let the numbers be a and b. Given that the product of the numbers is 3600.
ab = 3600
We can use the AM-GM Inequality here to solve the problem. The arithmetic mean-geometric mean (AM-GM) inequality states that the arithmetic mean of positive real numbers is greater than or equal to the geometric mean of the numbers.
AM-GM Inequality
Let a and b are positive numbers, arithmetic mean is greater than or equal to the geometric mean.
AM-GM Inequality states that:
or more in general for n numbers, the equation states that :
Now to answer the question: we known the product ab = 3600.
replace the product of the two numbers in the equation
we get:
The square root of 3600 is 60. So we get
Answer:
Hence the minimum value of the sum of the numbers is 120.
Proof
Proof for the AM-GM equation for two numbers:
Square both sides, we get
Expand and rearrange the terms, we get:
This equation holds true since the square of a number is always positive i.e
Let y = a-b
