GATE CS 1996 Q1.6 — Calculus: Limits, Continuity, Differentiability

To find the approximation for the second derivative $f''(x_0)$, we use the Taylor series expansions of $f(x_0 + h)$ and $f(x_0 - h)$ around the point $x_0$:

$$f(x_0 + h) = f(x_0) + h f'(x_0) + \frac{h^2}{2} f''(x_0) + \frac{h^3}{6} f'''(x_0) + \mathcal{O}(h^4) \quad \text{--- (1)}$$

$$f(x_0 - h) = f(x_0) - h f'(x_0) + \frac{h^2}{2} f''(x_0) - \frac{h^3}{6} f'''(x_0) + \mathcal{O}(h^4) \quad \text{--- (2)}$$

Adding equations (1) and (2) to eliminate the first derivative term $f'(x_0)$ gives:

$$f(x_0 + h) + f(x_0 - h) = 2f(x_0) + h^2 f''(x_0) + \mathcal{O}(h^4)$$

Rearranging the terms to solve for $f''(x_0)$:

$$h^2 f''(x_0) = f(x_0 + h) - 2f(x_0) + f(x_0 - h) - \mathcal{O}(h^4)$$

$$f''(x_0) \approx \frac{f(x_0 + h) - 2f(x_0) + f(x_0 - h)}{h^2}$$

This is the standard central difference formula for the second derivative.

Therefore, the correct option is D.