When 'a' + 'b' is given to be a constant, 'ab' has its maximum value when they are closest to each other, or if 'a' = 'b'. What happens when 'a' and 'b' are distinct?
Natural Numbers
Q.6: The sum of three distinct natural numbers is 25. What is the maximum value of their product?
Correct Answer
The maximum product is 560.
Detailed Solution
Let the three natural numbers be a, b and c. We know that a + b + c = 25.
Let us substitute some values and see where this is headed
a = 1, b = 2 c = 22, Product = 44
a = 2, b = 3 c = 20, product = 120
a = 5, b = 6 c = 14, product = 420
The numbers should be as close to each other as possible. (Just trial and error and one can figure this out). This property is an extension of the AM-GM Inequality.
25/3
∼ 8.
25 divided by 3 is approximately equal to 8. So, we should choose a, b, and c close enough to 8.
8 + 8 + 9 = 25. But the numbers need to be distinct. 8 * 7 * 10 come closest. The maximum product would be 560. Correct Answer : 560
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Inequalities are crucial to understand many topics that are tested in the CAT. Having a good foundation in this subject will make us tackling questions in Coordinate Geometry, Functions, and most importantly in Algebra much more comfortable.