Prove inequality

Dec 2015
975
128
Earth
Let \(\displaystyle a,b,c \geq 0 \;\) , \(\displaystyle \; a+b+c=1\) , Prove the following inequality :

$ 3\cos(\frac{1}{3})^{\cos(\frac{1}{3})^{\cos(\frac{1}{3})}}\geq\cos(a)^{\cos(b)^{\cos(c)}}+\cos(c)^{\cos(a)^{\cos(b)}}+\cos(b)^{\cos(c)^{\cos(a)}}\geq 2+\cos(1) $
 
Oct 2013
713
91
New York, USA
This is not a proof, but the inequality is the same concept that given a fixed perimeter, the largest rectangular area is a square. In this case there are three numbers instead of two and exponents instead of products. In other words, it's just a more complicated version of the fact that if a and b range from 0 to 1 and add to 1, the maximum value of ab is when a = b = 1/2. That handles the first two parts.
 
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