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Consider the function $\mathrm{f}:\left[\frac{1}{2}, 1\right] \rightarrow \mathrm{R}$ 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 $\mathrm{x}=\cos \frac{\pi}{12}$
Then
Only $(II)$ is correct
Both $(I)$ and $(II)$ are incorrect
Only$ (I)$ is correct
Both $(I)$ and $(II)$ are correct
Solution
$\mathrm{f}^{\prime}(\mathrm{x})=12 \sqrt{2} \mathrm{x}^2-3 \sqrt{2} \geq 0 \text { for }\left[\frac{1}{2}, 1\right]$
$\mathrm{f}\left(\frac{1}{2}\right)<0$
$\mathrm{f}(1)>0 \Rightarrow(\mathrm{A})$ is correct.
$f(x)=\sqrt{2}\left(4 x^3-3 x\right)-1=0$
Let $\cos \alpha=\mathrm{x}$,
$\cos 3 \alpha=\cos \frac{\pi}{4} \Rightarrow \alpha=\frac{\pi}{12}$
$\mathrm{x}=\cos \frac{\pi}{12}$
$(4)$ is correct.
