PROBLEM: Arithmetic Mean -- Geometric Mean Inequality

 

Find 5 different demonstrations (proofs) of the Arithmetic Mean -- Geometric Mean inequality for two non-negative numbers



One example.


Another suggestion.


Some Algebra.


Another example.


And still another example . . .


More algebra . . .


Using some geometry . . .


One More with Algebra


Using a family of curves xy = c and a line x + y = 2m


Can you generalize this theorem?

In the case of three positive quantities:

In general,

For a discussion of one proof of these generalizations, see Courant, R,. & Robbins, H. (1941) What is Mathematics? New York: Oxford University Press, pp. 361-365.


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