Answer to Question #181568 in Real Analysis for anuj

Question #181568

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.


1
Expert's answer
2021-05-12T03:48:02-0400

"(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\n\n\\\\[9pt]\n\nf''(x)=0\\le 2\n\n\\\\[9pt]\n\n|f^k(x)|\\le 2, k\\in (0,1,..,n-1) \n\\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.



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