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

Question

Consider the function $f:[\frac{1}{2},1]	o R$ 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

Accepted Answer

$$\begin{aligned}
& \mathrm{f}^{\prime}(\mathrm{x})=12 \sqrt{2} \mathrm{x}^2-3 \sqrt{2} \geq 0 \text { for }\left[\frac{1}{2}, 1\right] \
& \mathrm{f}\left(\frac{1}{2}\right)0 \Rightarrow(\mathrm{A})\end{aligned}$$ is correct.

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

Let $$\cos \alpha=\mathrm{x}$$,

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

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

(4) is correct.

Solutions · 1