If $x,y$ are real numbers with $x+y\geqslant 0$, determine the minimum value of the expression \[K=x^5+y^5-x^4y-xy^4+x^2+4x+7\]For which values of $x,y$ does $K$ take its minimum value?
Source: Cyprus 2022 Junior TST-1 Problem 3
Tags: algebra, inequalities
If $x,y$ are real numbers with $x+y\geqslant 0$, determine the minimum value of the expression \[K=x^5+y^5-x^4y-xy^4+x^2+4x+7\]For which values of $x,y$ does $K$ take its minimum value?