When does $\|p\|_k$ define a norm on $V$?

Question

Consider the real vector space $V$ of polynomials of degree less than or equal to $d$. For $p \in V$ define $$\lVert p\rVert_k = \mathrm {max} \{\lvert p(0)\rvert,\lvert p^{(1)}(0)\rvert,\dots,\lvert p^{(k)}(0)\rvert \},$$ where $p^{(i)}(0)$ is the $i^{\mathrm {th}}$ derivative of $p$ evaluated at $0$. Then $\lVert p\rVert_k$ defines a norm on $V$ if and only if

$1. \ $ $k \ge d-1$.

$2. \ $ $k < d$.

$3. \ $ $k \ge d$.

$4. \ $ $k < d-1$.

How can I proceed to solve this question? Please help me in this regard.

Thank you very much.

Answer 1

Hint. Note that $p(x)=x^d$ is non-zero polynomial of degree $d$ such that $p^{(k)}(0)=0$ for $k=0,1,\dots,d-1$. Then show that if $p(x)=\sum_{i=0}^d a_i x^i$ then $p^{(k)}(0)=k!a_k$ for $k=0,1,\dots,d$. Is there a non-zero polynomial $p$ of degree $\leq d$ such that $p^{(k)}(0)=0$ for $k=0,1,\dots,d$?