5th Row Coefficients, 6th Row Terms

x6+5x5y+10x4y2+kx3y3+10x2y4+5xy5+y60x^6+5x^5y+10x^4y^2+kx^3y^3+10x^2y^4+5xy^5+y^6\ge 0

Find the absolute value of the smallest possible kk such that the inequality above is true for all non-negative reals xx and y y .

Note: You may use the algebraic identities below.

  • (x+y)5=x5+5x4y+10x3y2+10x2y3+5xy4+y5(x+y)^5=x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5
  • (x+y)6=x6+6x5y+15x4y2+20x3y3+15x2y4+6xy5+y6(x+y)^6=x^6+6 x^5 y+15 x^4 y^2+20 x^3 y^3+15 x^2 y^4+6 x y^5+y^6

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