Say that functions $h$ and $y$ in $C(R)$ are such that $h(17)=1, h'(17)=0, y(17)=0, y'(17)=1$. Would $h$ and $y$ be linearly independent or dependent functions?
I believe that I should check to see if these could have linear combinations to where $h=y$ and $h'=y'$ but I'm not sure how to do that.
Suppose $ah+by=0$, which means that, for every $x\in\mathbb{R}$, $$ ah(x)+by(x)=0 $$ Then, also $ah'(x)+by'(x)=0$.
Evaluate both for $x=17$ and conclude that…