TITLE: why is my expectation of where the critical points are disagreeing with my graph?
QUESTION [0 upvotes]: given the parametrised function
$$f_t(x,y,z)=z^3+tz-x^2+y^2$$
We know that that the gradient vector field is 
$\nabla f_t=(-2x,2y,3z^2+t)$
Now I originally thought that the critical points were at $(0,0,\sqrt{\tfrac{-t}{3}})$. but when I graphed the level sets of the functions for t=0,1,-1. It seemed as though critical points happened at $(0,0,0), (0,0,\sqrt{\tfrac{1}{3}})(0,0,-\sqrt{\tfrac{1}{3}})$
where am i going wrong ?

REPLY [1 votes]: Note that 
$$t=1 \implies 3z^2+t >0 \quad \forall z$$
therefore we can't have a critical point for that value of the parameter $t$, more in general we need $t \le 0$ as a necessary condition to have critical points.
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