Maximum Of Two Variables' Formula $x$,$y$ are real numbers satisfying $(x-1)^{2}+4y^{2}=4$
find the maximum of $xy$ and justify it without calculus.
Does there exist a tricky solution using elementary inequalities (AM-GM or Cauchy-Schwarz) ?
I tried and got it's when $x=\dfrac{3+\sqrt{33}}{4}$
 A: If you need to justify only the solution without calculus, you can do it the following way: The solution you have is 
$$
x=\frac{1}{4} \left(3+\sqrt{33}\right) \qquad y=\frac{1}{4} \sqrt{\frac{1}{2} \left(15+\sqrt{33}\right)}
$$
If you go along the ellipse from that $(x,y)$ by a very small amount it will be along the direction 
$$
\Delta x = - 2 a y \qquad \Delta y = a (x-1) / 2
$$
for some very small $a$ (infinitesimal, so to say). Now compute 
$$
(x+\Delta x)(y + \Delta y) - x y = -\frac{1}{16} \left(\sqrt{33}-1\right) \sqrt{\frac{1}{2}
   \left(15+\sqrt{33}\right)} a^2,
$$
which is obviously negative, so the $x$ and $y$ you found must be a maximum.
Maybe, you can also find $x$ and $y$ in a similar way, by requiring that the linear term in the equation above (i.e. $(x+\Delta x)(y + \Delta y) - x y$) vanishes, but I haven't tried that.
EDIT: I tried it now, and it gives you the additional equation $-4y^2 + x(x-1)=0$, which you can combine with the original equation to easily find the solution first for $x$ and then for $y$.
A: If you set $y = k/x$, then you get a quartic equation in $x$ and you want to know the maximal $k$ such that there is a real solution for $x$. If you trace through all the complicated equations that define the solutions for a quartic equation, you should be able to figure it out. However I doubt this is the most efficient approach without calculus.
A: This is a circle of radius 2 centered at (1,0).   If you let x-1 = 2cosu and y = 2sinu then xy - (x-1)y + y  =4(cosu)(sinu) +2sinu = 2sin2u + 2 sinu = 2(sin2u + sinu).
Since sin2u < 0 for $\pi/2 < u <\pi$ and sinu = < 0 for $\pi < u < 2\pi$, the maximum is going to occur in the first quadrant.  Since both sinu and sin 2u are < 1 on 0 < u < $\pi/4$ the maximum must occur between $\pi/4$ and $\pi/2$.
Now let's look at the circle $x^2 + y^2$ = 1.  The points (x,y) on that circle are described as (cosu,sinu) so that xy = (cosu)(sinu) = (1/2)sin2u. Clearly sin2u is everywhere increasing in the first quadrant, so will take its maximum at 2u = $\pi$/2 or u = $\pi$/4.
This result has to be the same for all circles.  Going back to your original circle, choose the point on that circle where u = $pi$/4.  You are left only to compute the coordinates and do the multiplication.
