Is this a norm for $C^1[0,1]$?

Question

Let $C^1[0,1]$ be the space of differentiable functions in [0,1] in which their derivatives are continous in [0,1]. Define:

$ |||f|||_1=||f||_\infty =max \left \{ |f'(x)|: 0\leq x \leq 1\right \} $

Is $|||f|||_1 $ a norm in $C^1[0,1]$?

My Attempt to figure this out: My answer would be no because $ f(x)= c $ is a differentiable function in [0,1], its derivative is 0 and therefore $f'$ is continous. So for $|||f|||_1 $ to be a norm it needs have certain properties such as $|||f|||_1 = 0$ iff $f=0 $. But when f is a constant, its norm is $0$.

Is this correct? Am I missing something?

Answer 1

You are correct.When f=$0$, its norm is also 0.But not converse,failing the first property of norm.