Let $f:\mathbb{R}\rightarrow\mathbb{R}^n$ be a differentiable mapping with $f'(t)\neq\textbf{0}$ for all $t\in\mathbb{R}$, and let $\textbf{p}\in\mathbb{R}^n$ be a point not in $f(\mathbb{R})$.
(a) Show there is a point $\textbf{q}=f(t)$ on the curve $f(\mathbb{R})$ $\textit{closest}$ to $\textbf{p}$.
(b) Show that the vector $r=(\textbf{p}-\textbf{q})$ is orthogonal to the curve at $\textbf{q}$. $\textbf{Hint: }$Consider the function $t\mapsto|\textbf{p}-f(t)|$ and its derivative.
I'm not sure how to approach this. It appears to be a least squares problem. The hint on (b) seems to imply that the derivative of the distance formula is $0$ at $\textbf{q}$, but I'm not sure how that helps.
This is false. Let $$ p = (0,0) $$ and $$ f(t) = \left( \; e^{-t} \cos t , \; e^{-t} \sin t \; \right) $$