Exercise on equivalent norms I need to show that 2 norms are equivalent in this space: $X=C^{1}([a,b])$ .
The first norm is the standard $ ||f||_{1} $. The second norm is $ ||f||_{x} = |f(a)| +  ||f^{'}||_{\infty} $.
I can do one inequality, namely  $ ||f||_{1} < C||f||_{x} $,  could you please help me to find the opposite inequality?
Thanks a lot
PS: for the first part I used the fact that $f(t) = f(a) + \int_a^tf^{'}(\tau)d\tau$

Consider the sequence $(f_n)_n$ in $C^1([0,\pi])$ defined by
$$
f_n(x) = \sin(2nx).
$$
Note that
$$
\|f_n\|_1 = \int_0^\pi \lvert \sin(2nx) \rvert dx = \frac{1}{2n} \int_0^{2n\pi } \lvert \sin(x) \rvert dx
$$
Because $\lvert \sin(x) \rvert = \lvert \sin(x+\pi) \rvert$, we have 
$$
\|f_n\|_1 = \int_0^\pi \sin(x) dx = -\cos \pi + \cos(0)=2.
$$
But $f_n' = 2n \cos(2nx)$ so $\|f_n\|_x =2n$. This means $(f_n)_n$ is unbounded in $\|\cdot \|_x$ but bounded in $\|\cdot \|_1$, so the norms can't be equivalent.
