Norm majorization

Question

I have a task of finding out, if norms majorize themselves, what in my case is true if $\exists_{G>0} \forall_{f \in C^1([0,4])} \|f\|_i ≤ G\|f\|_j, i,j \in \{1,2\}$

1. $f$ is in $C^1([0,4])$
   $\|f\|_1 = |f(1)| + \max_{x\in[0,4]}|f'(x)|$
   $\|f\|_2 = \int_2^4 |f(t)|\,dt + \max_{x\in[0,4]}|f'(x)|$
2. $f$ is in $C^2([0,5])$
   $\|f\|_1 = \max_{x\in[0,5]}|f(x)| + \max_{x\in[0,5]}|f'(x)| + \max_{x\in[0,5]}|f''(x)|$
   $\|f\|_2 = |f(2)| + |f(4)| + \max_{x\in[0,5]}|f''(x)|$

I only managed to find out that in 2) $\|f\|_2 ≤2\|f\|_1$. Answers for other 3 cases or at least some tips how to find them would be really welcome.