I am supposed to find the points (x,y,z) satisfying the condition $x^2+2y^2-z^2-1=0$ that are the closest to origin (0,0,0). So basically, the idea was to find the minima of $\Lambda(x,y,z,\lambda) = \sqrt{x^2+y^2+z^2}+\lambda(x^2+2y^2-2z^2-1)$
For that I determined $\Lambda_x(x,y,z,\lambda) = \frac{x}{\sqrt{x^2+y^2+z^2}} + 2\lambda x$ $\Lambda_y(x,y,z,\lambda) = \frac{y}{\sqrt{x^2+y^2+z^2}} + 4\lambda y$ $\Lambda_z(x,y,z,\lambda) = \frac{z}{\sqrt{x^2+y^2+z^2}} - 2\lambda z$
But that would mean for $\lambda \neq 0$ $\lambda_1 = -\frac{1}{2\sqrt{x^2+y^2+z^2}}$ $\lambda_2 = -\frac{1}{4\sqrt{x^2+y^2+z^2}}$ $\lambda_3 = \frac{1}{2\sqrt{x^2+y^2+z^2}}$
Since $\lambda_1 \neq \lambda_2 \neq \lambda_3$ doesn't that mean that this function doesn't have any minima? I'm probably having a huge misunderstanding somewhere, but I just can't figure out where I went wrong, so please help me with this.