Find integer part of the maximum value of $x^4y+x^3y+x^2y+xy+xy^2+xy^3+xy^4$ for $x+y=3$ 
If $M$ is the maximum value of $x^4y+x^3y+x^2y+xy+xy^2+xy^3+xy^4$. Subject to $x+y=3$. Find the value of $\lfloor M \rfloor$.

I think the maximum of $M$ occurred when $x=y=3/2$. And I just put the value of that in M and found 34. What is the original answer with good solution ?? Somebody please help me.

After substitution $y=3-x$ we get:
$$M=-11x^4+66x^3-139x^2+120x$$
$$M'(x)=2(3-2x)(11x^2-33x+20),$$
which gives a maximal value $\frac{400}{11}$ for $x=\frac{3+\sqrt{\frac{19}{11}}}{2}$.
