Let x and y be positive real numbers. Prove that x^7 + y^7 >= x^4*y^3 + x^3*y^4

x^7 + y^7 - (x^4*y^3 + x^3*y^4)
= x^4(x^3-y^3) - y^4(x^3 - y^3)
= (x^4-y^4)(x^3-y^3)
= (x^2-y^2)(x^2+y^2) (x-y)(x^2 + xy + y^2)
= (x-y) (x+ y)(x^2+y^2) (x-y)(x^2 + xy + y^2)
= (x-y)^2 (x+ y)(x^2+y^2) (x^2 + xy + y^2)

as each term is non negative so
x^7 + y^7 - (x^4*y^3 + x^3*y^4) >= 0 ( > if x and y are not same and = if same)

or x^7 + y^7 >= x^4*y^3 + x^3*y^4