Limit, Continuity and Differentiability: JEE Main 2024 Mathematics question

Consider the function $f:\left[\frac{1}{2}, 1\right] \rightarrow \mathbb{R}$ defined by $f(x)=4 \sqrt{2} x^3-3 \sqrt{2} x-1$. Consider the statements (I) The curve $y=f(x)$ intersects the $x$-axis exactly at one point. (II) The curve $y=f(x)$ intersects the $x$-axis at $x=\cos \frac{\pi}{12}$. Then