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?