find the distance of point $P(0,0,1)$ from the level surface $f(x,y,z)=0$ of $f(x, y, z)=\left(z-x^{2}\right)(z+3 y)$

Question

given $$f(x, y, z)=\left(z-x^{2}\right)(z+3 y)$$

I am asking to find the distance of point $p=(0,0,1)$ from the level surface $f(x, y, z)= 0$.

The idea of what I am asked is pretty simple but How should I execute that?

Answer 1

Consider:

Objective function $x^2+y^2+(z-1)^2$, which is the square of the distance from the position $p$, and constrained to $(z-x^2)(z+3y)=0$.

So, your auxiliary function is $F=x^2+y^2+(z-1)^2+\lambda(z-x^2)(z+3y)$. Then the equations $$\frac{\partial F}{\partial x}=0,$$ $$\frac{\partial F}{\partial y}=0,$$ $$\frac{\partial F}{\partial z}=0,$$ together with $(z-x^2)(z+3y)=0$, going to give you where the distance to the square distance function: $$x^2+y^2+(z-1)^2,$$ reaches an extremum.