<s>[INST] If $x$ and $y$ are real numbers such that $4x^2 + y^2 = 4x - 2y + 7$, find the maximum value of $5x+6y$. If $x$ and $y$ are real numbers such that $4x^2 + y^2 = 4x - 2y + 7$, find the maximum value of $5x+6y$.
I did a little bit of manipulation and got $4(x+1)(x-2) + (y+1)^2=0$. I then got $x=2$ and $y=-1$ which means the maximum must be $4$, but the answer key says it's $16$. How come?
 [/INST] You've observed that $x\in[-1,2]$, but your mistake is that the maximum need not be at one of the endpoints. Once you know that 
$$4(x+1)(x-2) + (y+1)^2=0,$$
you see that any solution $(x,y)$ must have $-1\leq x\leq 2$. Then
$$y=-1\pm2\sqrt{-(x+1)(x-2)},$$
and so you want to find the maximum of
$$5x+6y=5x-1\pm2\sqrt{-(x+1)(x-2)},$$
for $x\in[-1,2]$, which I'm sure you can do by standard methods.
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