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 $g$.

Sketch: Suppose to reach a contradiction that $g$ has a maximum at $a\in U.$ Then $\nabla g (a)$ is the zero vector. Show that this implies the nonzero vector $f(a)$ is perpendicular to the vectors $\partial f/\partial x_k, k=1,\dots , n.$ But $Df(a)$ has rank $3,$ hence these vectors span $\mathbb {R}^3.$ This is a contradiction.
