GATE DA 2025 Q55: Nearest-mean classifier: is the decision function linear?

Consider a two-class problem in $\mathbb{R}^d$ with class labels red and green. Let $\mu_{red}$ and $\mu_{green}$ be the means of the two classes. Given test sample $x \in \mathbb{R}^d$, a classifier calculates the squared Euclidean distance (denoted by $\lVert \cdot \rVert^2$) between $x$ and the means of the two classes and assigns the class label that the sample $x$ is closest to. That is, the classifier computes

$$f(x) = \lVert \mu_{red} - x \rVert^2 - \lVert \mu_{green} - x \rVert^2$$

and assigns the label red to $x$ if $f(x) < 0$, and green otherwise. Which of the following statements is/are correct?