Let's Prove The AM-GM Inequality

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[DihinAmarasigha-uphf]

Ah...finally a great understandable proof for the Am Gm Inequality

[grrgrrgrr]

You can prove a much more general statement by first proving the inequality for weighted means of two elements. I.e.
t*a + (1-t)*b >= a^t*b^(1-t), with a, b>0 and t in (0, 1).
Fix t and b and let f(a) be the difference (which we wanna prove to be nonnegative). We first look at f' and find that f'(a) = t - t*a^(t-1)*b^(1-t). We see that f'(a)=0 at a=b. Furthermore, f' increases so f'(a)<0 at a<b and f'(a)>0 at a>b. This means that f itself takes a minimum at a=b. Clearly, f(b)=0, so the inequality of weighted means for two elements is proven.
To prove the inequality for weighted means for more than two elements, just apply the inequality for two elements multiple times.

[ΕκπαιδευτήριαΚαντά-ηω]

cube ID x^3+ψ^3+z^3>=3xψz if a=qubx b=qubψ z=qubz we proved identity cauchy for three qub means three root