Let a, b, c be three real number constants satisfying $a^2 + b^2 + c^2 \leq 1$. Define the function $f(x, y, z) = \frac{x^2 + y^2}{2(1+z)}$ under the constraints $(x-a)^2 + (y-b)^2 + (z-c)^2 \leq \mu^2$ and $x^2 + y^2 + z^2 \leq 1$. Now I want to calculate the minimal value of $f(x,y,z)$. My mathod is construct the following Laganrange:
$L(x, y, z, \lambda) = \frac{x^2 + y^2}{2(1+z)} + \lambda\left((x-a)^2 + (y-b)^2 + (z-c)^2 - \mu^2\right)$
Because the point $(x,y,z)$ is in the unit sphere, so I abandoned the constraint term that the point lies inside the unit sphere in the Lagrange function. After partial derivatives with respect to $x,y,z,\lambda$ and set them to zero to find critical points:
$L_x=\frac{x}{1+z}+2\lambda (x-a)=0$
$L_y=\frac{y}{1+z}+2\lambda (y-b)=0$
$L_z=-\frac{x^2+y^2}{2(1+z)^2}+2\lambda (z-c)=0$
$L_\lambda=(x-a)^2 + (y-b)^2 + (z-c)^2 - \mu^2=0$.
The question is how to solve the equations above?