Question

Show that if n is a natural number and α, β are real numbers with β > 0 then there exists a real function f with derivatives of all orders such that: (i) |f^(k)(x)| ≤ β for k ∈ {0, 1, ..., n − 1} and x ∈ (−∞, ∞); (ii) f^(k)(0) = 0 for k ∈ {0, 1, ..., n − 1}; (iii) f ⁽ⁿ⁾(0) = α.

Answer 1

$(i) |f^k(x)|\le \beta, k\in (0,1,..,n-1) \text{ and }x\in (-\infty,\infty)$

Take $f(x)=2$

   $|f(x)|=2\le 2$

$f'(x)=0\le 2 \\[9pt] f''(x)=0\le 2 \\[9pt] |f^k(x)|\le 2, k\in (0,1,..,n-1) \text{ and } x\in (-\infty,\infty)$

(ii) $f^k(0)=0, \text{ for } k\in (0,1,..,n-1)$

 Take $f(x)=0$

    $f^k(x)=0 \forall k\in (0,1,..,n-1$

   and $f(x)=x^n$

     Clearly $f^k(0)=0\forall k\in (0,1,..,n-1)$

(iii)$f^n(0)=\alpha$

   Take $f(x)=e^x$

  $f^0(0)=e^0=1$

$f'(x)=e^x$

$f'(0)=e^0=1$

Similarly $f^k(0)=1 \forall k\in (0,1,..,n-1)$

So, $f(x)=e^x$ is used here.