From S.L Linear Algebra:
Let $V$ be the vector space of functions which have derivatives of all orders, and let $D:V \rightarrow V$ be the derivative. What is the kernel of $D$?
This seems to be a very simple question, with very simple answer that I've constructed, though I'm not aware of how sufficient would it be.
I'm assuming, that "vector space of functions which have derivatives of all orders" is basically a function space of infinitely differentiable functions over some arbitrary field $K$ (since function space over a field $K$ is a vector space).
From basic calculus, we are aware of Fermat's interior extremum theorem, and that every point that has derivative of $0$ (stationary point) is local extremum (either maxima, minima or a saddle point).
Hence, the kernel of infinitely differentiable functions under $D$ is a space spanned by all local extremum's of infinitely differentiable functions.
Is this highly simple answer sufficient? or should I add something more.
Thank you!
I suppose that it is implicit here that $V$ is the space of all fonctions $f\colon\mathbb{R}\longrightarrow\mathbb R$ which have derivatives of all orders. Then$$\ker D=\left\{f\in V\,\middle|\,f'=0\right\}.$$But the functions $f$ such that $f'$ is the null function are the constant functions and only those functions.