Prove the Given Inequality
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Prove the Given Inequality

[From: ] [author: ] [Date: 12-08-20] [Hit: ]
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Given two positive numbers a and b, we define the geometric mean and arithmetic mean as follows:

Geometric Mean = sqrt{ab}
Arithmetic Mean = (a + b)/2

Prove that the following inequality is valid for all positive numbers a and b:

sqrt{ab} less than or equal to (a + b)/2

-
√ab ≤ (a+b)/2

ab ≤ (a²+2ab+b²) / 4

ab ≤ (a²+b²)/4 +ab/2

0 ≤ (a²+b²)/4 -ab/2

0 ≤ [(a/2-b/2)]²

where a and b are positive numbers

-
sqrt(ab) <= (a + b)/2
2sqrt(ab) <= a + b
4ab <= a^2 + b^2 + 2ab
0 <= a^2 + b^2 - 2ab
0 <= (a - b)^2

Proved as any square is greater or equal to zero
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