Functions Question 6
Question 6 - 2024 (29 Jan Shift 1)
Consider the function $f:\left[\frac{1}{2}, 1\right] \rightarrow R$ 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
(1) Only (II) is correct
(2) Both (I) and (II) are incorrect
(3) Only (I) is correct
(4) Both (I) and (II) are correct
Show Answer
Answer (4)
Solution
$f^{\prime}(x)=12 \sqrt{2} x^{2}-3 \sqrt{2} \geq 0$ for $\left[\frac{1}{2}, 1\right]$
$f\left(\frac{1}{2}\right)<0$
$f(1)>0 \Rightarrow(A)$ is correct.
$f(x)=\sqrt{2}\left(4 x^{3}-3 x\right)-1=0$
Let $\cos \alpha=x$,
$\cos 3 \alpha=\cos \frac{\pi}{4} \Rightarrow \alpha=\frac{\pi}{12}$
$x=\cos \frac{\pi}{12}$
(4) is correct.
