Equivalence of norms in $C^1[0,1]$

Question

i have the following problem/questions: I have to prove that $\lVert \cdot \rVert_1 \sim \lVert \cdot \rVert_{*} $ in $C^1[0,1]$; Where $\lVert \cdot \rVert_1$ is the usual $C^1[0,1]$ norm and $\lVert f \rVert_{*} =\lvert f(0)\rvert + \sup_{t\in[0,1]}\, \lvert f'(t) \rvert $. It's easy to show that $\lVert f \rVert_1 \geq \lVert f \rVert_{*} $. I have have some problem to check the other inequality. My first idea was to create a functional $T(f)=f(0)+f'(t)$ and to apply in some way the Banach-Steinhaus theorem. My second idea produces the following question: if i show that 2 different norms coincide on the norm-null function, are they equivalent? [$\lVert f \rVert_1 =0] \Rightarrow [\lVert f \rVert_{*}=0] $ and [$\lVert f \rVert_* =0] \Rightarrow [\lVert f \rVert_{1}=0$] implies that $\lVert \cdot \rVert_1 \sim \lVert \cdot \rVert_{*} $? Which idea do you suggest?