Kreyszig's FA Prob. $6$ section $2.8$

You need to consider $f$ with the following norm:

$$ ||f|| = \sup_{||x||=1}|f(x)| = \sup_{\max_{a \le t \le b}|x(t)|=1}|x'(c)| $$

where $x \in C^1[a, b]$.

Then you can take a non-negative $x_n' \in C[a, b]$ such that $x_n'(a)=x_n'(b)=0$, $x_n'(c) \rightarrow \infty$ and $x_n(b)=\int_a^b x'(s)ds=1$. For the existence of such functions check this.

You now have $|f(x_n)| \rightarrow \infty$.