If $x_0$ is extreme point of $f(x)$, why $\nabla f(x_0)$ is the normal vector of $z=f(x)$ ?
I am not sure whether it right. But when I try to understand why $\nabla F=\lambda \nabla G$ in lagrangian multiplier with constraint , my classmate tells me that $\nabla f(x_0)$ is the normal vector of $z=f(x)$ at $x_0$.
But I construct an example that $f(x)=x^2$, $x=0$ is a extreme point, but $\nabla f(0)=0$. I don't know what wrong with it.
This question is from my fuzzy with this question.
The correct statement should be:
The gradient $\nabla f$ of a function $f(x,y)$ is the normal vector of the level curve $f(x,y)=c$ at the point $(x,y)$.
Of course it could be a function of more than two variables.
It does not have to be at the extreme point. In fact, at the extreme point, the statement is not true since the gradient has no direction then.
So your example should be like this:
$$z=x^2+y^2, \nabla z=\langle 2x,2y\rangle$$
It is perpendicular to the level curve $x^2+y^2=c$ everywhere else except at the extreme point $(0,0)$.