Question

Consider the function $f :[\frac{1}{2},1]$⇢R defined by $f(x)=4\sqrt2x^3-3\sqrt2x-1$.Consider the statements
(1)The curve y=f(x) intersect the x-axis exactly at one point
(2)The curve y=f(x) intersect the x-axis at $x=cos\frac{\pi}{12}$
Then

• A. Only (II) is correct
• B. Both (I) and (II) are incorrect
• C. Only (I) is correct
• D. Both (I) and (II) are correct

Answer 1

Answer: (D)

Both (I) and (II) are correct

Answer 2

Step 1: Check the Derivative $f'(x)$ for Monotonicity

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

Step 2: Evaluate $f(x)$ at the Endpoints

$f\left(\frac{1}{2}\right) < 0$

$f(1) > 0$

Since $f(x)$ changes sign from negative to positive, there must be exactly one root in $\left[\frac{1}{2}, 1\right]$, confirming that statement (I) is correct.

Step 3: Check if $x = \cos \frac{\pi}{12}$ is a Root

Rewrite $f(x)$ in terms of $\cos \alpha$:

$f(x) = \sqrt{2}(4x^3 - 3x) - 1 = 0$

Let $\cos \alpha = x$, then $\cos 3\alpha = x$ gives $\alpha = \frac{\pi}{12}$, so:

$x = \cos \frac{\pi}{12}$

This confirms statement (II) is also correct.

So, the correct answer is: Both (I) and (II) are correct