1. **State the problem:** We are given the function $$y=\sqrt[3]{27-8x^3}$$ and asked to discuss its monotony, i.e., where it is increasing or decreasing. 2. **Recall the formula and rules:** To analyze monotony, we find the derivative $$y'$$ and determine where $$y' > 0$$ (increasing) or $$y' < 0$$ (decreasing). 3. **Rewrite the function:** $$y = (27 - 8x^3)^{\frac{1}{3}}$$ 4. **Find the derivative using the chain rule:** $$y' = \frac{1}{3}(27 - 8x^3)^{-\frac{2}{3}} \cdot (-24x^2) = -8x^2 (27 - 8x^3)^{-\frac{2}{3}}$$ 5. **Analyze the sign of $$y'$$:** - The term $$x^2 \geq 0$$ for all real $$x$$. - The term $$(27 - 8x^3)^{-\frac{2}{3}} > 0$$ for all $$x$$ where the expression inside the root is defined (since even root of a real number cubed is defined and the power is negative but even). - The negative sign in front makes $$y' \leq 0$$ for all $$x$$. 6. **Conclusion:** - $$y' = 0$$ only at $$x=0$$. - For $$x \neq 0$$, $$y' < 0$$, so the function is strictly decreasing everywhere except possibly flat at $$x=0$$. **Final answer:** The function $$y=\sqrt[3]{27-8x^3}$$ is monotonically decreasing on its entire domain.