Exercise 23, § 1

Functions of a Real Variable, II, 84

Let $\mathbf{f}$ be a vector function, continuously differentiable on an interval $[a, b] \subset \mathbf{R}$ , with values in a complete normed space.

a) Show that for $a \leq t \leq b$ one has

\mathbf{f}(t) = \frac{1}{b-a} \int_a^b \mathbf{f}(x)\, dx + \int_a^b \frac{x-a}{b-a} \mathbf{f}'(x)\, dx + \int_t^b \frac{x-b}{b-a} \mathbf{f}'(x)\, dx.

b) Deduce the inequalities

\| \mathbf{f}(t) \| \leq \frac{1}{b-a} \int_a^b \| \mathbf{f}(x) \| \, dx + \int_a^b \| \mathbf{f}'(x) \| \, dx

and

\| \mathbf{f}\left( \frac{1}{2}(a+b) \right) \| \leq \frac{1}{b-a} \int_a^b \| \mathbf{f}(x) \| \, dx + \frac{1}{2} \int_a^b \| \mathbf{f}'(x) \| \, dx.

Solution

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