I feel my IQ drops with every second I look at this excercise, that's why I ask you guys:

Given $\displaystyle C\subset \mathbb{R}^3$ with $\displaystyle (x,y,z)$ that satisfy $\displaystyle 2x^2+2y^2+z^2=1$ and $\displaystyle x=y^2+z^2$

Find $\displaystyle (x,y,z)\in C$ such that the distance to the origin is maximal/minimal

However, I can't find points that satisfy to both equations of $\displaystyle C$

We can derive

(1) $\displaystyle x\geq 0 $

(2) $\displaystyle y^2=1-2x^2-x\geq 0 $

(3) $\displaystyle z^2=2x^2+2x-1\geq 0 $

From (2) we get $\displaystyle x\in [0,\frac{1}{2}]$

From (3) we get $\displaystyle x\geq \frac{1}{2}$

Only $\displaystyle x= \frac{1}{2}$ seems ok. But it gives $\displaystyle y^2=z^2=0$, ...scheisse

So, I can't find any $\displaystyle x$ that could possibly satisfy the equations, let alone $\displaystyle y,z$

Can someone fix my brains? What's wrong here?