Thread: Obtaining the maximum of a shifted function

1. Obtaining the maximum of a shifted function

I have a continuous function f(x) , whose maximum Fm occuring at x=Xo is known.

Can we find the maximum of g(x) = f(x+A) + f(x-A) from the above information and the value of x where we have the maximum of g(x) ?

Is there any way to do this please let me know.

Thanks & Regards
Anindya

2. Is f differentiable? Over what interval are you maximizing the function? Does the maximum of f that you know occur at a critical point, or at an endpoint of your interval of interest?

3. Yes F is differentiable . Sorry i forgot to add that. The maximum of f occurs at a critical point

4. except in trivial cases (eg A=0), i dont think you can do this from the information provided. Do you have any more information about f(x)

5. No i don't.

6. Ok, f differentiable implies g is differentiable. The derivative of g is

$g'(x)=f'(x+A)+f'(x-A)$.

As SpringFan25 has noted, setting $g'(x)=0$ is a bit problematic. I'm not exactly sure how you would go about solving that. You know that $f'(x_{0})=0$. But I'm not seeing where you would go next. For, if you set $x+A=x_{0}$, then $x=x_{0}-A$, and $x-A=x_{0}-2A$. Then what? You don't know that the second term is zero.

7. Use the definition of derivative... Just and idea...

8. Originally Posted by Anindya
I have a continuous function f(x) , whose maximum Fm occuring at x=Xo is known.

Can we find the maximum of g(x) = f(x+A) + f(x-A) from the above information and the value of x where we have the maximum of g(x) ?

Is there any way to do this please let me know.

Thanks & Regards
Anindya
I know that the maximum of the function f(x+A) is x=Xo-A, and that the maximum of the function f(x-A) is x=Xo+A, but I'm not sure how to do it when the functions are added together. I'm not sure where to go on this one, like the previous poster said, is there anymore information about f(x)? I hope the facts above about the sperate functions help, otherwise I've just wasted space.