<s>[INST] Find the least value of the expression $x^2+2xy+2y^2+4y+7$ Find the least value of the expression
$x^2+2xy+2y^2+4y+7$
I am not able to solve this equation though i am able to differentiate it. 
 [/INST] The following one may be a more general solution without any trick. At first, let's recall a simple fact involving the quadric expression, which states that

$$ax^2+bx+c \geq \frac{4ac-b^2}{4a}~~~$$ 

holds for $a>0$ and with the equality
 holding iff $x=-\dfrac{b}{2a}.$
Hence, we have 
\begin{align*}
x^2+2y\cdot x+2y^2+4y+7 &\geq \frac{4 \cdot 1 \cdot (2y^2+4y+7)-(2y)^2}{4\cdot 1}\\&=y^2+4y+7\\&\geq \frac{4\cdot 1 \cdot 7-4^2}{4 \cdot 1}\\&=3,
\end{align*}
with the equality holding iff $x=-\dfrac{2y}{2 \cdot 1}$ and $y=-\dfrac{4}{2\cdot 1}$, namely, $x=2, y=-2.$
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