How to identify family of function from given property
A={$f:Rto R$}
with following properties
f has derivative of every order
$forall x,yin R$ $f(x+y)-f(y-x)=2xf'(y)$
then I have to find whole family
Till now I just able to show $f(x)-f(-x)/2x=2xf'(0)$
I wanted to solve this problem.
Please provide just hint..
Thanks in Advanced
real-analysis derivatives
asked Nov 27 at 13:28
MathLover
45710
add a comment |
A={$f:Rto R$}
with following properties
f has derivative of every order
$forall x,yin R$ $f(x+y)-f(y-x)=2xf'(y)$
then I have to find whole family
Till now I just able to show $f(x)-f(-x)/2x=2xf'(0)$
I wanted to solve this problem.
Please provide just hint..
Thanks in Advanced
real-analysis derivatives
asked Nov 27 at 13:28
MathLover
45710
Plug $y = x$ into the equation and use Taylor series ($f$ has a Taylor series since it's infinitely differentiable).
– Rchn
Nov 27 at 13:46
Sir I have done that , I got f(y)=f(0)+yf'(y/2).
– MathLover
Nov 27 at 13:53
form here how I one identify given family is polynomial of degree less than 2
– MathLover
Nov 27 at 13:54
add a comment |
A={$f:Rto R$}
with following properties
f has derivative of every order
$forall x,yin R$ $f(x+y)-f(y-x)=2xf'(y)$
then I have to find whole family
Till now I just able to show $f(x)-f(-x)/2x=2xf'(0)$
I wanted to solve this problem.
Please provide just hint..
Thanks in Advanced
real-analysis derivatives
asked Nov 27 at 13:28
MathLover
45710
A={$f:Rto R$}
with following properties
f has derivative of every order
$forall x,yin R$ $f(x+y)-f(y-x)=2xf'(y)$
then I have to find whole family
Till now I just able to show $f(x)-f(-x)/2x=2xf'(0)$
I wanted to solve this problem.
Please provide just hint..
Thanks in Advanced
real-analysis derivatives
real-analysis derivatives
asked Nov 27 at 13:28
MathLover
45710
asked Nov 27 at 13:28
MathLover
45710
asked Nov 27 at 13:28
MathLover
45710
asked Nov 27 at 13:28
MathLover
45710
asked Nov 27 at 13:28
MathLover
45710
45710
Plug $y = x$ into the equation and use Taylor series ($f$ has a Taylor series since it's infinitely differentiable).
– Rchn
Nov 27 at 13:46
Sir I have done that , I got f(y)=f(0)+yf'(y/2).
– MathLover
Nov 27 at 13:53
form here how I one identify given family is polynomial of degree less than 2
– MathLover
Nov 27 at 13:54
add a comment |
Plug $y = x$ into the equation and use Taylor series ($f$ has a Taylor series since it's infinitely differentiable).
– Rchn
Nov 27 at 13:46
Sir I have done that , I got f(y)=f(0)+yf'(y/2).
– MathLover
Nov 27 at 13:53
form here how I one identify given family is polynomial of degree less than 2
– MathLover
Nov 27 at 13:54
Plug $y = x$ into the equation and use Taylor series ($f$ has a Taylor series since it's infinitely differentiable).
– Rchn
Nov 27 at 13:46
Plug $y = x$ into the equation and use Taylor series ($f$ has a Taylor series since it's infinitely differentiable).
– Rchn
Nov 27 at 13:46
Sir I have done that , I got f(y)=f(0)+yf'(y/2).
– MathLover
Nov 27 at 13:53
Sir I have done that , I got f(y)=f(0)+yf'(y/2).
– MathLover
Nov 27 at 13:53
form here how I one identify given family is polynomial of degree less than 2
– MathLover
Nov 27 at 13:54
form here how I one identify given family is polynomial of degree less than 2
– MathLover
Nov 27 at 13:54
add a comment |
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Plug $y = x$ into the equation and use Taylor series ($f$ has a Taylor series since it's infinitely differentiable).
– Rchn
Nov 27 at 13:46
Sir I have done that , I got f(y)=f(0)+yf'(y/2).
– MathLover
Nov 27 at 13:53
form here how I one identify given family is polynomial of degree less than 2
– MathLover
Nov 27 at 13:54