Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be continuously differentiable such that $x^{*}=\text{argmin}_{x\in\mathbb{R}^{n}}f(x)$ and gradient dominated, i.e. \begin{align*} f(x) - f(x^{*}) \leq \tau||\nabla f(x)||_{2}^{2}, \quad \forall x \in \mathbb{R}^{n}, \end{align*} for some $\tau>0$. Find such a $f$ that is non-convex.
Remark: The class gradient dominated functions was originally introduced by Boris Polyak. Here, they say that such functions need not be convex.
Edit: \begin{align*} f(x) = \begin{cases} x^{2} & \text{if } x \leq 1,\\ -\frac{1}{2}x^{2}+3x-\frac{3}{2} & \text{if } 1 \leq x \leq 2,\\ x^{2}-3x-\frac{9}{2} & \text{if } 2 \leq x. \end{cases} \end{align*} is such a function with e.g. $\tau=3$.