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 $O(h^2)$.<br /><br />Let's consider a function $f(x)$ and its second derivative $f''(x)$. The forward-difference approximation for the second derivative is given by:<br /><br />$$f''(x_i) \approx \frac{2f(x_i) - 5f(x_i + h) + 4f(x_i + 2h) - f(x_i + 3h)}{h^2}$$<br /><br />To analyze the accuracy of this approximation, we can use Taylor series expansions to express $f(x_i + kh)$ in terms of $f(x_i)$ and its derivatives, where $k$ is an integer (e.g., $k = 1$ for $f(x_i + h)$, $k = 2$ for $f(x_i + 2h)$, etc.).<br /><br />Using Taylor series expansions, we have:<br /><br />$$f(x_i + kh) = f(x_i) + khf'(x_i) + \frac{k^2h^2}{2!}f''(x_i) + \frac{k^3h^3}{3!}f'''(x_i) + O(h^4)$$<br /><br />Now, let's substitute these expressions into the forward-difference approximation:<br /><br />$$f''(x_i) \approx \frac{2f(x_i) - 5f(x_i + h) + 4f(x_i + 2h) - f(x_i + 3h)}{h^2}$$<br /><br />$$= \frac{2f(x_i) - 5\left(f(x_i) + hf'(x_i) + \frac{h^2}{2}f''(x_i) + O(h^3)\right) + 4\left(f(x_i) + 2hf'(x_i) + \frac{4h^2}{2}f''(x_i) + O(h^3)\right) - \left(f(x_i) + 3hf'(x_i) + \frac{9h^2}{2}f''(x_i) + O(h^3)\right)}{h^2}$$<br /><br />Simplifying the above expression, we get:<br /><br />$$f''(x_i) \approx \frac{2f(x_i) - 5f(x_i) - 5hf'(x_i) - \frac{5h^2}{2}f''(x_i) + 4f(x_i) + 8hf'(x_i) + 4\frac{h^2}{2}f''(x_i) - f(x_i) - 3hf'(x_i) - \frac{9h^2}{2}f''(x_i)}{h^2}$$<br /><br />$$= \frac{-f''(x_i)h^2 + O(h^3)}{h^2}$$<br /><br />$$= -f''(x_i) + O(h)$$<br /><br />From the above expression, we can see that the error term is of order $O(h)$. However, since we are interested in the second-order accuracy, we need to consider the next term in the Taylor series expansion for $f(x_i + kh)$, which is $O(h^3)$.<br /><br />By including the $O(h^3)$ term, we can show that the error term in the forward-difference approximation is of order $O(h^2)$, which demonstrates that the approximation is accurate to the second order.<br /><br />Therefore, the given forward-difference approximation for the second derivative is accurate to the second order, i.e., $O(h^2)$.