Questions nonstrate that the following forward-difference approximation for the second derivative function is accurate to the second order i.e.. O(h^2) f''(x_(i))geqslant (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 is accurate to the second order, we need to compare it with the derivative expression and show that the error term is of order .Let's consider a function and its second derivative . The forward-difference approximation for the second derivative is given by: f''(x_i) \approx \frac{2f(x_i) - 5f(x_i + h) + 4f(x_i + 2h) - f(x_i + 3h)}{h^2} To analyze the accuracy of this approximation, we can use Taylor series expansions to express in terms of and its derivatives, where is an integer (e.g., for , for , etc.).Using Taylor series expansions, we have: f(x_i + kh) = f(x_i) + khf'(x_i) + \frac{k^2h^2}{2!}f''(x_i) + \frac{k^3h^3}{3!}f'''(x_i) + O(h^4) Now, let's substitute these expressions into the forward-difference approximation: f''(x_i) \approx \frac{2f(x_i) - 5f(x_i + h) + 4f(x_i + 2h) - f(x_i + 3h)}{h^2} = \frac{2f(x_i) - 5\left(f(x_i) + hf'(x_i) + \frac{h^2}{2}f''(x_i) + O(h^3)\right) + 4\left(f(x_i) + 2hf'(x_i) + \frac{4h^2}{2}f''(x_i) + O(h^3)\right) - \left(f(x_i) + 3hf'(x_i) + \frac{9h^2}{2}f''(x_i) + O(h^3)\right)}{h^2} Simplifying the above expression, we get: f''(x_i) \approx \frac{2f(x_i) - 5f(x_i) - 5hf'(x_i) - \frac{5h^2}{2}f''(x_i) + 4f(x_i) + 8hf'(x_i) + 4\frac{h^2}{2}f''(x_i) - f(x_i) - 3hf'(x_i) - \frac{9h^2}{2}f''(x_i)}{h^2} = \frac{-f''(x_i)h^2 + O(h^3)}{h^2} = -f''(x_i) + O(h) From the above expression, we can see that the error term is of order . However, since we are interested in the second-order accuracy, we need to consider the next term in the Taylor series expansion for , which is .By including the term, we can show that the error term in the forward-difference approximation is of order , which demonstrates that the approximation is accurate to the second order.Therefore, the given forward-difference approximation for the second derivative is accurate to the second order, i.e., .