Prove that: (is true for any positive x,y):? #x^x*y^y>=((x+y)/2)^(x+y)#

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#x^x*y^y>=((x+y)/2)^(x+y)#

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Answer 1 · 2018-02-27T14:51:05

See below.

Consider #f(x) = x ln x#

This function has a convex hypograph because

#f''(x) = 1/x > 0#

so in this case

#f((x+y)/2) le 1/2(f(x)+f(y))# or

#((x+y)/2)ln((x+y)/2) le 1/2( x ln x + y ln y)# or

#((x+y)/2)^((x+y)/2) le (x^x y^y)^(1/2)#

and finally squaring both sides

#((x+y)/2)^(x+y) le x^x y^y#