Finding minima and maxima to $f(x,y) = x^2 + x(y^2 - 1)$ in the area $x^2 + y^2 \leq 1$ I'm asked to find minima and maxima on the function 
$$f(x,y) = x^2 + x(y^2 - 1)$$
in the area 
$$x^2 + y^2 \leq 1.$$
My solution:
$$\nabla (f) = (2x + y^2 - 1, 2xy)$$
Finding stationary points 
$$2xy = 0$$
$$2x + y^2 - 1 = 0$$
gets me $(0,1),(0,-1),(\frac{1}{2},0)$.
Stuyding the boundary: 
$$x^2 + y^2 = 1$$
$$y^2 = 1 - x^2 $$
$$f(x,y) = f(x, 1-x^2) = x^2 -x^3$$
Finding stationary points on the boundary:
$$f'(x,y) = 2x - 3x^2 = 0$$
gives $(0, 1), (0,-1), (\frac{2}{3}, \sqrt{\frac{5}{9}}),(\frac{2}{3}, -\sqrt{\frac{5}{9}})$.
So in total i've got the stationary points 
$$(\frac{1}{2} , 0) | (0,1)| (0,-1)| (\frac{2}{3},\sqrt{\frac{5}{9}})|(\frac{2}{3}, -\sqrt{\frac{5}{9}}).$$
which give the function values of: 
$$-\frac{1}{4}, 0,0,0.15,0.15$$.
which gives the minima: $-\frac{1}{4}$ and maxima $0.15$.
The minima is correct but the maxima should be 2. Why? What stationary point am I missing?

HINT: the searched maximum is given by $2$ and will be attained for $$x=-1,y=0$$
