1. **Problem Statement:** Find the second derivative $f''(3)$ of the function $f(x) = x^4$ at $x=3$ using the forward difference operator with given data points $x = 0, 3, 6, 9, 12, 15$. 2. **Formula for the second derivative using forward difference operator:** $$f''(x) = \frac{1}{h^2} \left[ \Delta^2 f(x) - \frac{2}{2} \Delta^3 f(x) + \frac{11}{12} \Delta^4 f(x) \right]$$ where $h$ is the step size and $\Delta^n f(x)$ are the forward differences. 3. **Calculate $h$:** Given points are equally spaced by $h = 3$ since $3 - 0 = 3$. 4. **Calculate $f(x)$ values at given points:** $f(0) = 0^4 = 0$ $f(3) = 3^4 = 81$ $f(6) = 6^4 = 1296$ $f(9) = 9^4 = 6561$ $f(12) = 12^4 = 20736$ $f(15) = 15^4 = 50625$ 5. **Calculate forward differences:** - First differences $\Delta f(x)$: $\Delta f(0) = f(3) - f(0) = 81 - 0 = 81$ $\Delta f(3) = f(6) - f(3) = 1296 - 81 = 1215$ $\Delta f(6) = f(9) - f(6) = 6561 - 1296 = 5265$ $\Delta f(9) = f(12) - f(9) = 20736 - 6561 = 14175$ $\Delta f(12) = f(15) - f(12) = 50625 - 20736 = 29889$ - Second differences $\Delta^2 f(x)$: $\Delta^2 f(0) = \Delta f(3) - \Delta f(0) = 1215 - 81 = 1134$ $\Delta^2 f(3) = 5265 - 1215 = 4050$ $\Delta^2 f(6) = 14175 - 5265 = 8910$ $\Delta^2 f(9) = 29889 - 14175 = 15714$ - Third differences $\Delta^3 f(x)$: $\Delta^3 f(0) = \Delta^2 f(3) - \Delta^2 f(0) = 4050 - 1134 = 2916$ $\Delta^3 f(3) = 8910 - 4050 = 4860$ $\Delta^3 f(6) = 15714 - 8910 = 6804$ - Fourth differences $\Delta^4 f(x)$: $\Delta^4 f(0) = \Delta^3 f(3) - \Delta^3 f(0) = 4860 - 2916 = 1944$ $\Delta^4 f(3) = 6804 - 4860 = 1944$ 6. **Apply the formula at $x=3$:** $$f''(3) = \frac{1}{3^2} \left[ \Delta^2 f(3) - 1 \cdot \Delta^3 f(3) + \frac{11}{12} \Delta^4 f(3) \right] = \frac{1}{9} \left[ 4050 - 4860 + \frac{11}{12} \times 1944 \right]$$ Calculate inside the bracket: $$4050 - 4860 = -810$$ $$\frac{11}{12} \times 1944 = 11 \times 162 = 1782$$ Sum: $$-810 + 1782 = 972$$ Therefore: $$f''(3) = \frac{972}{9} = 108$$ 7. **Verification:** The exact second derivative of $f(x) = x^4$ is $f''(x) = 12x^2$. At $x=3$, $f''(3) = 12 \times 9 = 108$, which matches our numerical result. **Final answer:** $$\boxed{108}$$