Exercise 12, § 3

a) Let $f$ be a vector function defined and continuous on the finite interval $I = [-a, +a]$, taking its values in a normed space $E$ over $\mathbf{R}$ and twice differentiable on $I$. If one puts $M_0 = \sup_{x \in I} \|f(x)\|, \quad M_2 = \sup_{x \in I} \|f''(x)\|$, show that for all $x \in I$ one has

(express each of the differences $f(a) - f(x), \quad f(-a) - f(x)$).

b) Deduce from a) that if $f$ is a twice differentiable function on an interval $I$ (bounded or not), and if $M_0 = \sup_{x \in I} \|f(x)\|$ and $M_2 = \sup_{x \in I} \|f''(x)\|$ are finite, then so is $M_1 = \sup_{x \in I} \|f'(x)\|$, and one has:

Show that in these two inequalities the numbers 2 and $\sqrt{2}$ respectively cannot be replaced by smaller numbers (consider first the case where one supposes merely that $f$ admits a second right derivative, and show that in this case the two terms of the preceding inequalities can become equal, taking for $f$ a real function equal "in pieces" to second degree polynomials).

c) Deduce from b) that if $f$ is $p$ times differentiable on $\mathbf{R}$, and if $M_p = \sup_{x \in \mathbf{R}} \|f^{(p)}(x)\|$ and $M_0 = \sup_{x \in \mathbf{R}} \|f(x)\|$ are finite, then each of the numbers $M_k = \sup_{x \in \mathbf{R}} \|f^{(k)}(x)\|$ is finite (for $1 \leq k \leq p-1$) and