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QUESTION
(Sipnayan 2011 High School Elims, Average Question 2) Suppose x and y are real numbers such that 2x^2+y^2-2xy+12y+72\le0. What is the value of x^2y?
(Sipnayan 2011 High School Elims, Average Question 2) Suppose x and y are real numbers such that 2x^2+y^2-2xy+12y+72\le0. What is the value of x^2y?
Overview/Laconic Solution Sketch
Group into squares and apply the Trivial Inequality.
Discussion
It's evident that Cauchy-Schwarz and AM-GM isn't very viable. We'll have to stick with good old Trivial Inequality (x^2\ge 0, \forall x \in \mathbb{R}) (\forall is the universal quantifier)
The first sensible thing to do would be to try to group the left hand side into a sum of squares. If we tried guessing from the term 2xy, it doesn't bring us far: \left(x^2-2xy+y^2\right) + x^2 + 12y + 72The more annoying term turns out to be 12y and 72. Now note that 72 isn't a square, but rather half a square. Better yet, 2\cdot 72 = 144 = 12^2. Hence, 12y+72 = \left( y+12 \right)^2 -\frac{y^2}{2}. Now we'll find that y^2/2 is just what the neighboring expression needs.
Solution
\begin{eqnarray*} 2x^{2}+y^{2}-2xy+12y+72 & = & 2x^{2}-2xy+\frac{y^{2}}{2}+\frac{y^{2}}{2}+12y+72\\ & = & \frac{1}{2}\left(2x-y\right)^{2}+\frac{1}{2}\left(y+12\right)^{2}\\& \le&0 \end{eqnarray*}which implies y=-12 and x=-6. We have x^2y=\boxed{-432}
Group into squares and apply the Trivial Inequality.
Discussion
It's evident that Cauchy-Schwarz and AM-GM isn't very viable. We'll have to stick with good old Trivial Inequality (x^2\ge 0, \forall x \in \mathbb{R}) (\forall is the universal quantifier)
The first sensible thing to do would be to try to group the left hand side into a sum of squares. If we tried guessing from the term 2xy, it doesn't bring us far: \left(x^2-2xy+y^2\right) + x^2 + 12y + 72The more annoying term turns out to be 12y and 72. Now note that 72 isn't a square, but rather half a square. Better yet, 2\cdot 72 = 144 = 12^2. Hence, 12y+72 = \left( y+12 \right)^2 -\frac{y^2}{2}. Now we'll find that y^2/2 is just what the neighboring expression needs.
Solution
\begin{eqnarray*} 2x^{2}+y^{2}-2xy+12y+72 & = & 2x^{2}-2xy+\frac{y^{2}}{2}+\frac{y^{2}}{2}+12y+72\\ & = & \frac{1}{2}\left(2x-y\right)^{2}+\frac{1}{2}\left(y+12\right)^{2}\\& \le&0 \end{eqnarray*}which implies y=-12 and x=-6. We have x^2y=\boxed{-432}
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