gradient of implicit function

Question

$f : U \subset R^3 \to R$ is differentiable function.

$f^-(0)$ is regular surface.

Is gradient f(p) $\neq$ 0 for at all $p \in f^-(0)$ ?

I think the above question is true.
By the way, I do not know where to start the proof. Give me a hint.

Answer 1

Is not true. Take for example the following function $$ f(x_1, x_2, x_3) := ( |x|^2 - 1 )^2, $$ where $|x| = \sqrt{x_1^2 + x_2^2 + x_3^2}$.

Clearly $f^{-1}(0)$ is the unit sphere but you can check that the gradient is zero everywhere on each point of the sphere, because each point is a minimum point.