Exercise 13, § 3

a) Let $f$ be a twice differentiable real function on $\mathbf{R}$, such that $(f(x))^2 \leq a$ and $(f'(x))^2 + (f''(x))^2 \leq b$ on $\mathbf{R}$; show that

on $\mathbf{R}$ (argue by contradiction, noting that if the function $f^2 + {f'}^2$ takes a value $c > \max(a, b)$ at a point $x_0$ then there exist two points $x_1, x_2$ such that $x_1 < x_0 < x_2$ and that at $x_1$ and $x_2$ the function $f'$ takes values small enough that $f^2 + {f'}^2$ takes values $< c$; then consider a point of $[x_1, x_2]$ where $f^2 + {f'}^2$ attains its supremum on this interval).

b) Let $f$ be a real function $n$ times differentiable on $\mathbf{R}$, and such that $(f(x))^2 \leq a$ and $(f^{(n-1)}(x))^2 + (f^{(n)}(x))^2 \leq b$ on $\mathbf{R}$; show that then

on $\mathbf{R}$ for $1 \leq k \leq n$. (Argue by induction on $n$; note that, by exerc. 12 the supremum $c$ of $(f'(x))^2$ on $\mathbf{R}$ is finite; show that one necessarily has $c \leq \max(a, b)$ by reducing to a contradiction: assuming that $c > \max(a, b)$ choose the constants $\lambda$ and $\mu$ so that for the function $g = \lambda f + \mu$ one has $|g(x)| \leq 1,\ |g'(x)| \leq 1$, yet one cannot have $(g(x))^2 + (g'(x))^2 \leq 1$ for all $x$.)