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Question: The arithmetic mean of two numbers a and b is defined as (a+b)/2; the geometric mean of two positive numbers a and b is defined as √ab.
(a) For two positive numbers, which of the two means is larger? Justify your answer. [Hint: Define f(x)=(a+x)/2 - √ax for fixed a.]
(b) For three positive numbers a, b, c, the arithmetic and geometric mean are (a + b + c)/3 and √3 abc, respectively. Which of the two means of three numbers is larger?
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