Optimization with an ellipse I have worked on this one for a while and I can not make my answer match the author's.
Find the points on the ellipse $4x^2 + y^2 = 4$ that are the farthest away from the point (1,0).
I have:
$$4x^2 + y^2 = 4$$
and then the distance formula, so I set y to terms of x and I get
$$\sqrt{(x-1)^2 + (2-2x)^2}$$
Setting the difference the a square this gives me a derivative of 
$$10x-10$$
which gives me a zero of 1, this is wrong according to the book and I am not sure why.

We want to maximize $\sqrt{(x-1)^2+(y-0)^2}$, given that $4x^2+y^2=4$.
Equivalently, we want to maximize $(x-1)^2+y^2$, same side condition.
But $y^2=4-4x^2$. 
So we want to maximize $(x-1)^2+(4-4x^2)$. Your turn.
