<s>[INST] Minimal distance between the origin and intersection of $x^2 = 2yz$ and $x^2+3y^2+2z^2 = 30$ 
Find the minimal distance between the origin and intersection of $x^2 = 2yz$ and $x^2+3y^2+2z^2 = 30$

Attempt:
$2yz+3y^2+2z^2 = 30$
I called $f(y,z) = 2yz+3y^2+2z^2$
Then, $\nabla f(y,z) = (2z+6y,2y+4z) = 0 \Rightarrow y=z=0$
So the critical point is $(0,0,0)$, which is clearly a minimum. But, I think this is wrong, because the minimum point is not necessarily the minimal distance to the origin. How do I get the closest point to the origin?
 [/INST] i have $$\sqrt{x^2+y^2+z^2}=f(x,y,z)$$ for $$x^2$$ we have $$x^2=2yz$$ and further
$$2z^2=30-x^2-3y^2=2yz-3/2y^2$$ we get
$$z^2=15-yz-\frac{3}{2}y^2$$ and we have to optimize
$$\sqrt{yz-\frac{1}{2}y^2+15}$$
you can consider the function $$h(y,z)=yz+15-\frac{1}{2}y^2$$ and compute the partial derivatives with respect to $$y,z$$
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