1. **Problem statement:** We want to find a root of the function $f(x) = x^3 - 4x^2 - 10$ in the interval $[1,2]$ using fixed point iteration. 2. **Fixed point iteration method:** We rewrite the equation $f(x) = 0$ as $x = g(x)$ for some function $g(x)$. 3. **Rewrite the equation:** Starting from $x^3 - 4x^2 - 10 = 0$, we isolate $x$: $$x^3 = 4x^2 + 10$$ One possible rearrangement is: $$x = \sqrt[3]{4x^2 + 10}$$ So we define: $$g(x) = \sqrt[3]{4x^2 + 10}$$ 4. **Iteration formula:** $$x_{n+1} = g(x_n) = \sqrt[3]{4x_n^2 + 10}$$ 5. **Check convergence condition:** For fixed point iteration to converge, $|g'(x)| < 1$ near the root. Calculate derivative: $$g'(x) = \frac{1}{3}(4x^2 + 10)^{-2/3} \cdot 8x = \frac{8x}{3(4x^2 + 10)^{2/3}}$$ Evaluate roughly in $[1,2]$ to ensure $|g'(x)| < 1$. 6. **Perform iterations starting with an initial guess $x_0 = 1.5$ (midpoint of interval):** - $x_1 = g(1.5) = \sqrt[3]{4(1.5)^2 + 10} = \sqrt[3]{4 \times 2.25 + 10} = \sqrt[3]{9 + 10} = \sqrt[3]{19} \approx 2.668$ - $x_2 = g(2.668) = \sqrt[3]{4(2.668)^2 + 10} = \sqrt[3]{4 \times 7.117 + 10} = \sqrt[3]{28.468 + 10} = \sqrt[3]{38.468} \approx 3.37$ The values are increasing and moving away from the interval, indicating divergence. 7. **Try another rearrangement:** Rewrite $f(x) = 0$ as: $$x = \frac{x^3 - 10}{4x}$$ Define: $$g(x) = \frac{x^3 - 10}{4x}$$ 8. **Check derivative:** $$g'(x) = \frac{(3x^2)(4x) - (x^3 - 10)(4)}{(4x)^2} = \frac{12x^3 - 4x^3 + 40}{16x^2} = \frac{8x^3 + 40}{16x^2} = \frac{8x^3}{16x^2} + \frac{40}{16x^2} = \frac{x}{2} + \frac{5}{2x^2}$$ Evaluate at $x=1.5$: $$g'(1.5) = \frac{1.5}{2} + \frac{5}{2 \times (1.5)^2} = 0.75 + \frac{5}{4.5} \approx 0.75 + 1.111 = 1.861 > 1$$ So this also does not guarantee convergence. 9. **Conclusion:** Fixed point iteration may not converge easily for this function on $[1,2]$ with simple rearrangements. 10. **Alternative:** Use other root-finding methods like Newton-Raphson or bisection for better convergence. **Final answer:** Fixed point iteration with $g(x) = \sqrt[3]{4x^2 + 10}$ diverges on $[1,2]$. No suitable $g(x)$ found here for guaranteed convergence in this interval.