Consider the function $f : [\frac{1}{2}, 1] \to ℝ$ 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 $x = \cos \frac{π}{12}$ . Then

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Only (II) is correct.

Both (I) and (II) are correct.

Both (I) and (II) are incorrect.

Only (I) is correct.

answer is B.

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Detailed Solution

$\begin{array}{l} f (x) = 4 \sqrt{2} x^{3} - 3 \sqrt{2} x - 1 \\ f^{'} (x) = 12 \sqrt{2} x^{2} - 3 \sqrt{2} = 0 \\ x = \frac{1}{2} O N L Y \Rightarrow S t a t e m e n t 1 i s TRUE \end{array}$

$\begin{array}{l} f (x) = 4 \sqrt{2} x^{3} - 3 \sqrt{2} x - 1 = 0 \Rightarrow 4 x^{3} - 3 x = \frac{1}{\sqrt{2}} \\ ∴ x = \cos \frac{π}{12} i s a r o o t \Rightarrow S t a t e m e n t 2 i s t r u e \end{array}$