the following claim true or false: $\bigtriangledown f(x)$ always points in the same direction as $x-\hat{x}$ if $f(\hat{x})=0$

Question

Take $f: \mathbb{R}^{n}\rightarrow \mathbb{R}, f\geq0 \ \forall x \in \mathbb{R}^{n}$. A statement in an exercise claims the following:

given an $x' \in \mathbb{R}^{n}$, let $\hat{x}$ be the point closest to $x'$ satisfying $f(\hat{x})=0.$ if $f$ is continuously differentiable, then the gradient $\bigtriangledown f(\hat{x})$ always points in the same direction as $x' - \hat{x}$.

This appears wrong to me as if $f \geq 0\ \forall x$, then $\bigtriangledown f(\hat{x}) = 0$. And even if we removed the condition $f \geq 0 \ \forall x$, taking $n=1, f=x$ with $x'=-1$ and $\hat{x}=0$ is a simple counter example. Am I missing something?

Answer 1

You are right, and your counterexamples show it. The claim in the exercise is false.

Answer 2

I suspect that the claim might be a typo, and that it's talking about $\nabla (x')$ instead of $\nabla f (\hat{x})$. And "point in the same direction" is clearly false if taken literally, but suppose instead it meant "lie in the same half-space" (i.e., have positive dot product).

Even so, you'd need more conditions on $f$ to make this true. I might suggest throwing out that text.