Norm of a linear functional, sup, definite integral I'm trying to find the norm of this mapping:
$$\phi: C^1([0,1]) \ni f \rightarrow \int_0^{1/2} f(t)dt + f'(\frac{2}{3}) \in \mathbb{R}$$
with $||f|| = \sup_{t \in [0,1]}|f(t)| + \sup_{t \in [0,1]}|f'(t)|$
So far, I've proven that this functional is bounded:
$|\int_0^{1/2} f(t)dt + f'(\frac{2}{3}) | \le \frac{1}{2} \sup _{t \in [0,1]} |f(t)| + \sup _{t \in [0,1]} |f'(t)| $
$\frac{1}{2} \sup _{t \in [0,1]} |f(t)| $ is due to the Mean value theorem
So for sure  $||\phi|| \le 1$.
How can I find the optimal constant?

Take the sequence $(f_n)$ defined by
$f_n(x)=(1/n)sin(n(x-(2/3))$,
then
$\|f_n \|= 1/n +1; \space \space \space \space -1/n +1 \leq \phi (f_n) \leq 1/n +1$
So
$\phi (f_n /(\|f_n\|)) \to 1$.
