Q:

Using Lagrange's Mean Value Theorem to prove equality of norms

I'm looking for a proof using Lagrange's Mean Value Theorem of the following:
Prove that in $C^1[0,1]$ vector-space,
$\left\|f\right\|=\left|f(0)\right|+\left\|f'\right\|_\infty$ norm is equivalent with $\left\|f\right\|_\infty+\left\|f'\right\|_\infty$.
I'm trying to use the following inequality:
$$
\alpha(\left\|f\right\|_\infty + \left\|f'\right\|_\infty) \leq \left|f(0)\right|+\left\|f'\right\|_\infty \leq \beta(\left\|f\right\|_\infty + \left\|f'\right\|_\infty); \beta = 1
$$

A:

If $x\in[0,1]$, 
$$|f(x)-f(0)|\leq x\sup_{y\in[0,x]}|f'(y)|\leq x\|f'\|_\infty$$
by Lagrange's theorem.
Hence
$$|f(x)|\leq|f(0)|+x\|f'\|_\infty\leq |f(0)|+\|f'\|_\infty$$
so, taking the supremum as $x$ varies in $[0,1]$, you get
$$\|f\|_\infty\leq |f(0)|+\|f'\|_\infty$$
so
$$\|f\|_\infty+\|f'\|_\infty\leq |f(0)|+2\|f'\|_\infty\leq 2|f(0)|+2\|f'\|_\infty$$
hence $\alpha=1/2$ works.

