Demonstrate that the following forward-difference approximation for the second derivative of a function is accurate to the second order,ie., O(h^2) f''(x_(i))cong (2f(x_(i))-5f(x_(i)+h)+4f(x_(i)+2h)-f(x_(i)+3h))/(h^2)

To demonstrate that the given forward-difference approximation for the second derivative of a function is accurate to the second order, we need to compare it with the Taylor series expansion of the second derivative.Let's consider a function that is twice differentiable. We can expand in a Taylor series around as follows: where is an integer and is the step size.Now, let's substitute into the above equation to get the expressions for , , , and : Now, let's substitute these expressions into the given forward-difference approximation: Substituting the expressions for , , , and , we get: Simplifying the above expression, we get: