1. **Stating the problem:** Given the function $$Y = \frac{6}{5} x^5 - \frac{2}{7} x^6$$, we want to analyze it, find its critical points, and understand its behavior. 2. **Formula and rules:** To find critical points, we use the derivative $$Y' = \frac{dY}{dx}$$ and set it equal to zero. Critical points occur where $$Y' = 0$$ or where $$Y'$$ is undefined. 3. **Find the derivative:** $$Y' = \frac{d}{dx} \left( \frac{6}{5} x^5 - \frac{2}{7} x^6 \right) = \frac{6}{5} \cdot 5 x^{4} - \frac{2}{7} \cdot 6 x^{5} = 6 x^{4} - \frac{12}{7} x^{5}$$ 4. **Set the derivative equal to zero:** $$6 x^{4} - \frac{12}{7} x^{5} = 0$$ 5. **Factor out common terms:** $$x^{4} \left(6 - \frac{12}{7} x \right) = 0$$ 6. **Solve for $$x$$:** - Either $$x^{4} = 0 \Rightarrow x = 0$$ - Or $$6 - \frac{12}{7} x = 0 \Rightarrow 6 = \frac{12}{7} x \Rightarrow x = \frac{6 \times 7}{12} = \frac{42}{12} = \frac{7}{2} = 3.5$$ 7. **Interpretation:** The critical points are at $$x = 0$$ and $$x = 3.5$$. 8. **Summary:** The function $$Y = \frac{6}{5} x^5 - \frac{2}{7} x^6$$ has critical points at $$x=0$$ and $$x=3.5$$ where the slope of the curve is zero, indicating potential maxima, minima, or points of inflection. **Final answer:** Critical points at $$x=0$$ and $$x=3.5$$.