Find the points on the hyperbolic cylinder $x^2 - z^2 - 1 = 0$ that are closest to the origin.
Attempt:
$f(x,y,z) = x^2+y^2+z^2$ subject to constraint $x^2-z^2-1=0$. Then $z^2 = x^2-1.$
$h(x,y) = x^2+y^2+(x^2-1) = 2x^2 + y^2 - 1$; then $h_x = 4x = 0$ and $h_y = 2y = 0$.
But (0,0) is incorrect as there are no points on the cylinder where both x and y are zero.
The book claims that because the First Derivative Test found the point in the domain of h where h has a minimum value, as opposed to the point on the cylinder where h has a minimum value.
Then it goes on to say that we can resolve this problem if we treat y and z as independent variables.
So my question is: What is the reasoning behind setting y and z as independent variables? What does changing independent variables to y and z do? I'd like to get some intuition on this subject.
hint: consider the Lagrange-function $$f(x,y,z,\lambda)=x^2+y^2+z^2+\lambda(x^2-z^2-1)$$