is $\|p\| = \max |p'(x)|$ norm?

Question

Let $X$ be the the space of polynomials on $[0, 1]$ and let $\|p\| = \max|p'(x)|$, where $p'$ is the derivative of p. Is $\| . \|$ a norm on $X$?

1- clearly the norm is non-negative

2- let $\lambda \in \mathbb{R}$, then $\|\lambda p\| = \max|\lambda p'(x)|= |\lambda |\max|p'(x)|$

3- let $p,q \in X$, $\|p+q\|=\max |p'(x)+q'(x)|\le \max |p'(x)|+\max|q'(x)| = \|p\|+\|q\|$

Answer 1

No, because $1\neq0$, but $\lVert1\rVert=0$.

Answer 2

No, it is not. What is $\Vert 1\Vert$?