You say that has 4 continuous derivatives but then ask about g?

Also, does the problem really say "4 continuous derivatives"? This is true of any function that has2continuous functions. Also can we take it that "i" is just a variable and not such that ?

Finally, where did that "h" come from? Are you sure the final equation shouldn't be (f(i+h)- 2f(i)+ f(i- h))/h^2= f''(i)+ O(h^2) ??

Write f(x) as a Taylor's polynomial of degree 2, about x= i, and do the algebra to find the left side of that equation.