1. **State the problem:** We have the function $$f(x) = 1 - 2kx + kx^2 - x^3$$ and we want to find the range of values for the parameter $$k$$ such that $$f(x)$$ is decreasing on its entire domain. 2. **Recall the rule for decreasing functions:** A function is decreasing on its entire domain if its derivative is less than or equal to zero for all $$x$$. 3. **Find the derivative:** $$f'(x) = \frac{d}{dx}(1 - 2kx + kx^2 - x^3) = -2k + 2kx - 3x^2$$ 4. **Rewrite the derivative:** $$f'(x) = -2k + 2kx - 3x^2 = 2kx - 2k - 3x^2$$ 5. **Condition for decreasing:** We want $$f'(x) \leq 0$$ for all $$x$$. 6. **Analyze the quadratic in $$x$$:** Rewrite $$f'(x)$$ as: $$f'(x) = -3x^2 + 2kx - 2k$$ Since the leading coefficient $$-3 < 0$$, the parabola opens downward. 7. **For $$f'(x) \leq 0$$ everywhere, the quadratic must be non-positive for all $$x$$. This happens if the quadratic has no real roots or a single root (discriminant $$\leq 0$$). 8. **Calculate the discriminant $$\Delta$$:** $$\Delta = (2k)^2 - 4(-3)(-2k) = 4k^2 - 24k = 4k(k - 6)$$ 9. **Set discriminant condition:** $$4k(k - 6) \leq 0$$ Divide both sides by 4 (positive, so inequality direction stays): $$k(k - 6) \leq 0$$ 10. **Solve inequality:** The product $$k(k - 6) \leq 0$$ means $$k$$ is between 0 and 6 inclusive: $$0 \leq k \leq 6$$ 11. **Conclusion:** The function $$f(x)$$ is decreasing on its entire domain if and only if $$k$$ satisfies: $$0 \leq k \leq 6$$ **Final answer:** B) $$0 \leq k \leq 6$$