We are given
1= a+ b = a^{a+b} + b^{a+b}
So 1- (a^ab^b + a^b b^a)
= a^{a+b} + b^{a+b} - (a^ab^b + a^b b^a)
= a^a(a^b-b^b) + b^a(b^b-a^b) = (a^a - b^a)(a^b - b^b)
For a > b both the terms are non -ve so we have and if b > a then both terms are -ve and hence above is positive
1- (a^ab^b + a^b b^a) >=0 and hence the result
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