Consider the function f : [1/2, 1] → R defined by f(x) = 4√2x^3 − 3√2x − 1.

Question

Consider the function \({f}:\left[\frac{1}{2}, 1\right] \rightarrow \mathrm{R}\) 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 \(x=\cos \frac{\pi}{12}\)

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

Answer 1

Correct option is (4) Both (I) and (II) are correct

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

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

\(\mathrm{f}(1)>0 \Rightarrow(\mathrm{A})\) is correct.

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

Let \(\cos \alpha=\mathrm{x}\)

\(\cos 3 \alpha=\cos \frac{\pi}{4} \Rightarrow \alpha=\frac{\pi}{12}\)

\(\mathrm{x}=\cos \frac{\pi}{12}\)