1. State the problem: Find the x-intercept(s) of the function $$y = x^4 - 4x^3 + 10$$, i.e., find values of $$x$$ such that $$y=0$$. 2. Set the function equal to zero: $$x^4 - 4x^3 + 10 = 0$$ 3. Attempt to find real roots by inspection or factorization. Try substituting simple values: - At $$x=0$$: $$0 - 0 + 10 = 10 \neq 0$$ - At $$x=1$$: $$1 - 4 + 10 = 7 \neq 0$$ - At $$x=2$$: $$16 - 32 + 10 = -6 \neq 0$$ 4. Since the polynomial is quartic, try to find critical points by derivative to check behavior: $$y' = 4x^3 - 12x^2$$ 5. Factor derivative: $$y' = 4x^2 (x - 3)$$ Critical points at $$x=0$$ and $$x=3$$ 6. Evaluate function at critical points: - At $$x=0$$: $$y=10$$ - At $$x=3$$: $$81 - 108 + 10 = -17$$ 7. Since $$y(3) < 0$$ and $$y(0) > 0$$, the function crosses the x-axis between $$x=2$$ and $$x=3$$. 8. Try to find exact roots by solving $$x^4 - 4x^3 + 10=0$$. Check discriminant or attempt substitution. 9. Attempt substitution $$z = x - 1$$ or try to write it as a quadratic in $$x^2$$, but no easy factorization appears. 10. Use the Rational Root Theorem possibilities: factors of 10 over factors of 1, i.e., $$\pm 1, \pm 2, \pm 5, \pm 10$$. Test these values all yield non-zero. 11. Since no rational roots, approximate numerically roots between 2 and 3 using Intermediate Value Theorem. 12. Approximately root near $$x \approx 2.7$$ where function crosses x-axis. 13. Because the polynomial degree is even and leading coefficient positive, it tends to positive infinity at $$\pm\infty$$, suggesting two real roots or zero. 14. Graphing or numerical methods confirm two real roots near 1.35 and 2.65, but since polynomial is not factorable with simple methods, conclude x-intercepts are approximately $$\boxed{x \approx 1.35, x \approx 2.65}$$. Final answer: The x-intercepts of $$y = x^4 - 4x^3 + 10$$ are approximately at $$x \approx 1.35$$ and $$x \approx 2.65$$.