9. If the positive real numbers $\mathbf{x}$ and $\mathbf{y}$ satisfy $x^{3}+y^{3}=(4 x-5 y) y$, then the maximum value of $\mathbf{y}$ is
【Answer】 $\frac{1}{3}$
【Solution】【Detailed】
Let $x=k y$, then $y=\frac{4 k-5}{k^{3}+1}$. Let $f(k)=\frac{4 k-5}{k^{3}+1}$, then $f^{\prime}(k)=\frac{(k-2)\left(8 k^{2}+k+2\right)}{\left(k^{3}+1\right)^{2}}$, hence $f(k) \leq f(2)=\frac{1}{3}$, so the maximum value of $y$ is $\frac{1}{3}$.
Therefore, the answer is $\frac{1}{3}$.