Taylor series for 6th order second derivative method

Question

The problem I'm working is

Using Taylor expansions, derive a sixth order method for approximating the second derivative of a given sufficiently smooth function f(x).

Is the only way to do this, using Taylor Expansions, to use 7 points (x, x+h, x-h, ...) and then combine the Taylor Expansions f(x), f(x+h), f(x-h), ... in some way? If so, I haven't been able to find a way to combine them to cancel out the derivative terms I need to have cancel. If not, how else can I use Taylor Expansions to solve this?

Thanks!

Answer 1

You can simplify by starting from $$ \frac{f(x-kh)-2f(x)+f(x+kh)}{(kh)^2}=f''(x)+\frac1{12}f^{(4)}(kh)^2+\frac1{360}f^{(6)}(kh)^4+O(h^6) $$ for $k=1,2,3$ so that you only have to solve a $3×3$ system.