Question 155
Let $\mathcal{F}$ be the set of continuous functions $f:[0,1] \rightarrow \mathbb{R}$ satisfying the condition $\max_{0 \leq x \leq 1}|f(x)|=1$, and let $I: \mathcal{F} \rightarrow \mathbb{R}$,
$$
I(f)=\int_{0}^{1} f(x) \, \mathrm{d}x - f(0) + f(1)
$$
a. Prove that $I(f) \lt 3$ for every $f \in \mathcal{F}$.
b. Determine $\sup \{I(f) \mid f \in \mathcal{F}\}$.
$$
I(f)=\int_{0}^{1} f(x) \, \mathrm{d}x - f(0) + f(1)
$$
a. Prove that $I(f) \lt 3$ for every $f \in \mathcal{F}$.
b. Determine $\sup \{I(f) \mid f \in \mathcal{F}\}$.