Given the function $f(x) = \ln(1+x) - x + \frac{1}{2}x^2 - kx^3$, where $0 < k < \frac{1}{3}$. (1) Prove: $f(x)$ has a unique extremum point and a unique zero point on the interval $(0, +\infty)$; (2) Let $x_1, x_2$ be the extremum point and zero point of $f(x)$ on the interval $(0, +\infty)$ respectively. (i) Let $g(t) = f(x_1 + t) - f(x_1 - t)$. Prove: $g(t)$ is monotonically decreasing on the interval $(0, x_1)$; (ii) Compare the sizes of $2x_1$ and $x_2$, and prove your conclusion.
Given the function $f(x) = \ln(1+x) - x + \frac{1}{2}x^2 - kx^3$, where $0 < k < \frac{1}{3}$.\\
(1) Prove: $f(x)$ has a unique extremum point and a unique zero point on the interval $(0, +\infty)$;\\
(2) Let $x_1, x_2$ be the extremum point and zero point of $f(x)$ on the interval $(0, +\infty)$ respectively.\\
(i) Let $g(t) = f(x_1 + t) - f(x_1 - t)$. Prove: $g(t)$ is monotonically decreasing on the interval $(0, x_1)$;\\
(ii) Compare the sizes of $2x_1$ and $x_2$, and prove your conclusion.