Find min & max of $f(x,y) = x + y + x^2 + y^2$ when $x^2 + y^2 = 1$ Problem: Find the maximum and minimal value of $f(x,y) = x + y + x^2 + y^2$ when $x^2 + y^2 = 1$.
Since $x^2 > x$ (edit $x^2 \geq x$) for all $x \in \mathbb{R}$, $f$ is bowl-ish with a minimal value in the bottom.
This is a critical point which means that we can set partial derivatives of $f$ equal to $0$ and try to solve for $x$ and $y$ 
$\nabla f = (1+2x, 1+2y) = (0,0) \implies (x,y) = (\frac{-1}{2}, \frac{-1}{2})$
So we get the minimal value $f(\frac{-1}{2}, \frac{-1}{2}) = \frac{-1}{2} + \frac{-1}{2} + (\frac{-1}{2})^2 \frac{-1}{2})^2 = -\frac{1}{2}$
But how about the maximal value? How does $x^2 + y^2 = 1$ restrict $f$? 

Your method to get the minimum gets in the entire plane, but you want the minimum on the unit circle $x^2+y^2=1$.
You can use Lagrange multipliers. You could also parameterize the unit circle on one variable then find the extrema on that variable. Two parametrizations are
$$x=\cos\theta,\quad y=\sin\theta,\quad 0\le\theta<2\pi$$
or
$$y=\pm\sqrt{1-x^2},\quad -1\le x\le 1$$
In your problem it looks like the cosine/sine parametrization would be easier.
