x,y,z are non-negative reals that satisfy x+y+z=1.The maximun value of X^3Y^3+Y^3Z^3+Z^3X^3 has the form a/b where a and b are coprime integers .what is the value of a+b?

Question

x,y,z are non-negative reals that satisfy x+y+z=1.The maximun value  of X^3Y^3+Y^3Z^3+Z^3X^3 has the form a/b where a and b are coprime integers .what is the value of a+b?

experimentx · Answer

http://www.wolframalpha.com/input/?i=max [{x^3y^3%2By^3z^3%2Bz^3x^3%2C+x%2By%2Bz%3D%3D1%2C+x%3E0%2C+y%3E0%2C+z%3E0}%2C+{x%2C+y%2C+z}]

experimentx · Answer

65

experimentx · Answer

try rearrangement equality ... 
x^6+y^6+z^6 >= X^3Y^3+Y^3Z^3+Z^3X^3
or try Lagrange multipliers

anonymous · Answer

a+b=65?

experimentx · Answer

i guess ... that's WA tells me.

anonymous · Answer

does langrange multiplier gives the exact value

experimentx · Answer

i guess ... it does. Ugly method though.

experimentx · Answer

try something like this http://math.stackexchange.com/questions/184029/inequality-frac116abcd3-geq-abcbcdcdadab

experimentx · Answer

rearrangement does not work

anonymous · Answer

thanks ,i got it