I have a function f::$\mathbb{R^2} \rightarrow \mathbb{R}$, I know it is C¹ and given a point $P \in \mathbb{R^2}$, $f(P) = 0$. I have to show $\nabla f(P) = (0, 0)$.
Now, I was able to show that C¹ implies differentiablity as I thought I would need that, but it seems to me I need more data to make this proof. As I understand f has a critical point in point P $\iff \nabla f(P) = (0, 0)$ so I need to show, ONLY knowing f is C¹ and that $f(P) = 0$ that P is a critical point, but I think with only these things it can't be achieved.
Am I right? Any help will be appreciated.
Looks like I answered the question in a comment, so just to make sure the question doesn't appear unanswered, I'll copy it here.
You are right. It's not enough to know that $f$ is $C^1$ and $f(P) = 0$ to conclude that $\nabla f(P) = 0$. As a counterexample, take $f(x,y) = x$. Then $f(0,0) = 0$ but $\nabla f(0,0) = (1,0)$.