19.2.1.4 Properties of normed spaces

Problem: On \( C^{1}[0 ; 1] \) the following norms are given: \[ \begin{array}{l} \|f\|_{1}=|f(0)|+\int_{0}^{1} t^{2}\left|f^{\prime}(t)\right| d t \\ \|f\|_{2}=\int_{0}^{1}|f(t)| d t+\int_{0}^{1} t^{2}\left|f^{\prime}(t)\right| d t . \end{array} \] Is it true that one of these norms is stronger than the other one? Are they equivalent?