<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] Also, we can use C-S here:
$$19.5=\sqrt{6.5^2\cdot9}=\sqrt{(2.5^2+6^2)((2x-1)^2+(y+1)^2)}\geq$$
$$\geq2.5(2x-1)+6(y+1)=5x+6y+3.5,$$
which gives $$5x+6y\leq16.$$
The equality occurs for $$(2.5,6)||(2x-1,y+1),$$ which says that $16$ is a maximal value.
</s>