Consider the function f : [ 1 2 , 1 ] R defined by f ( x ) = 4 2 x 3 - 3 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 = cos12

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Solution

Q.
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

1 Both (I) and (II) are incorrect.

2 Only (I) is correct.

3 Both (I) and (II) are correct.

4 Only (II) is correct.

Ans.
(3)

$We have, f (x) = 4 \sqrt{2} x^{3} - 3 \sqrt{2} x - 1$

$So, f' (x) = 12 \sqrt{2} x^{2} - 3 \sqrt{2} \geq 0 for x \in [\frac{1}{2}, 1]$

$So, f (x) is increasing.$

$Now, f (\frac{1}{2}) < 0 and f (1) > 0$

$∴ f (x) intersects x -axis at exactly one point.$

$So, statement-I is correct.$

$Let \cos α = x$

$∴ \sqrt{2} (4 \cos^{3} α - 3 \cos α) - 1 = 0$

$\Rightarrow \sqrt{2} \cos 3 α = 1 \Rightarrow \cos 3 α = \frac{1}{\sqrt{2}} \Rightarrow 3 α = \frac{π}{4}$

$\Rightarrow α = \frac{π}{12}$

$So, statement-II is correct.$

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