Equivalance of norms Let $X$ be the vector space of all real valued functions defined on $[0,1]$ having continuous first-order derivatives. How to show that the following norms are equivalent:
$\|f\|_1 = |f(0)| + \|f'\|_\infty$, $\|f\|_2 = \|f\|_\infty + \|f'\|_\infty$ where $\|f\|_\infty$ is the usual sup norm on the space of continuous functions. 

Since clearly $\|f\|_1\le \|f\|_2$ for all $f\in X$, we need only find some $c$ such that $\|f\|_2\le c\|f\|_1$ for all $f\in X$.
In fact $c=3$ works: Let $a=\|f\|_1$ and $x\in (0,1]$. Then by the IVT, we have 
$$f(x)=f(0)+(x-0)\cdot f'(y)$$
for some $y\in (0,x)$. From $f(0)\le a$, $f'(y)\le a$, and $0<x\le 1$ we conclude $f(x)\le 2a$, similarly $f(x)\ge -2a$. Ultimately, $\|f\|_2\le 2a+a=3\|f\|_1$.
