mathoverflow_347178

Let $f : \mathbb R^n \to \mathbb R, n \geq 2$ be a $C^1$ function. Is it true that $$\sup_{x \in \mathbb R^n}f(x) = \sup_{x\in \mathbb R^n} f(x+\nabla f(x))$$?

theorem mathoverflow_347178 :
    answer(sorry) ↔ ∀ᵉ (n ≥ 2) (f : ℝ^n → ℝ) (hf : ContDiff ℝ 1 f),
        (BddAbove (range f) ↔ BddAbove (range (fun x ↦ f (x + gradient f x)))) ∧
        (⨆ x, (f x : EReal)) = ⨆ x, (f (x + gradient f x) : EReal)

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