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. 

$\| f\|_1 \leq C \| f \|_2$ seems obvious because a pointwise value must be less than the sup.
For the other direction, my hint is that
$$\|f\|_{\infty} - |f(0)| = \sup_{x \in [0,1]} |f(x)| - |f(0)| \leq  \sup_{x \in [0,1]} |f(x) - f(0)|$$
Now use the fundamental theorem of calculus and some obvious inequalities.
