<s>[INST] 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.
 [/INST] $x^{4}y+x^{3}y+x^{2}y+xy+xy^{2}+xy^{3}+xy^{4}=xy+xy(x+y)+xy(x^{2}+y^{2})+xy(x^{3}+y^{3})=xy+3xy+xy((x+y)^{2}-2xy)+xy((x+y)^{3}-3x^{2}y-3xy^{2})=4xy+xy(9-2xy)+xy(27-3xy(x+y))=13xy-2(xy)^{2}+xy(27-9xy)=40xy-11(xy)^{2}.$
Now take $xy=k$ and you have $k\leq \frac{9}{4}$. Now our expression is $40k-11k^{2}$. Using derivative calculate $k$ such that the expression is maximum and check that the value of $k$ is less than $\frac{9}{4}$
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