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