9. Let the function $f(x)=\sqrt[3]{1+x}-\lambda x$, where $\lambda>0$.
(1) Find the range of $\lambda$ such that the function $f(x)$ is monotonically decreasing on $[0,+\infty)$;
(2) Can this monotonicity be extended to the entire domain $(-\infty,+\infty)$?
(3) Solve the inequality $2 x-\sqrt[3]{1+x}<12$.
9. Solution: (1) Let $0 \leqslant x_{1}3$.
$\because f\left(x_{1}\right)-f\left(x_{2}\right)>0, \therefore \lambda>\frac{1}{M}, \because M>3, \therefore \frac{1}{M}0$, $g(x)$ is a monotonically increasing function. $\because g(7)=12$, $\therefore 2 x-\sqrt[3]{1+x}<12 \Leftrightarrow g(x)<g(7), \therefore x<7, \therefore$ the solution set is $(-\infty, 7)$.