Critical point of a function - $\Bbb R^n$ Analysis Consider $f=(f_1,f_2,f_3): U \rightarrow \mathbb{R}^3$ a function not identically null, $f\in C^1$ and rank $3$ at every point of the open $U \subset \mathbb{R}^n$, $n \geq 3$. Show that $g(x)= f_1^2(x)+f_2^2(x)+f_3^2(x)$, $x \in U$, hasn't maximum in $U$.
Suggestion: Suppose by contradiction considering $\nabla g$; look the sign of $g$.

There is a 'geometry' intuition behind your question. Observe that $g$ is the composition of $f$ and the square of the distance to the origen in $\mathbb{R}^3$. Namely, let $s : \mathbb{R}^3 \to \mathbb{R}$ be $s(x,y,z) := x^2 + y^2 + z^2$ be the square of the distance to $(0,0,0)$. Then $g = s \circ f$. Since $f$ has rank 3 at any point $x_0$ then there are vectors $v$ at $x_0$ such that the derivative of $f$ at $x_0$ takes $v$ to a vector $df(v)$ at $f(x_0)$ pointing outside of the sphere whose radius is $g(x_0)$. So $x_0$ can not be a maximum of $g$.
