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
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
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√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
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
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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
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