Suppose that $f:\mathbb{R}^n \rightarrow\mathbb{R}$ is given by $f(x)=a_1{x_1}^2+a_2{x_2}^2+...+a_n{x_n}^2$, where $x=(x_1,x_2,...,x_n)$ and at least one $a_j$ is not zero. Then we can conclude that
$f$ is not everywhere differentiable
The gradient $(\nabla{f})(x)\neq{0}$ for every $x\in\mathbb{R}^n$
If $x\in\mathbb{R}^n$ is such that $(\nabla{f})(x)={0}$ then f(x)=0
If $x\in\mathbb{R}^n$ is such that f(x)=0 then $(\nabla{f})(x)={0}$
1$\rightarrow$ false since $(Df)(x)=[2a_1x_1,2a_2x_2,...,2a_nx_n]$ which exist everywhere
2$\rightarrow$ false since $(\nabla{f})(x)=2a_1x_1e_1+2a_2x_2e_2+...+2a_nx_ne_n$ which is $0$ only if $x=0$
3$\rightarrow$ true
4$\rightarrow$true since $f(x)=0$ implies $a_1{x_1}^2+a_2{x_2}^2+...+a_n{x_n}^2=0$, let $a_1\neq{0}$ implies $x_1=a_2=...=a_n=0$ hence $(\nabla{f})(x)=2a_1x_1e_1+2a_2x_2e_2+...+2a_nx_ne_n=0$
is my explanation is correct?