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

MCQ

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

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

$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.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If five $G.M.’s$ are inserted between $486$ and $2/3$ then fourth $G.M.$ will be

$\tan \left( {{{90}^o} - {{\cot }^{ - 1}}\frac{1}{3}} \right) = $

In which of the following graphs $x = c$ is the point of inflection .

If ${\log _{\sqrt 3 }}\left( {\frac{{|z{|^2} - |z| + 1}}{{2 + |z|}}} \right) < 2$, then the locus of $z$ is

Corner points of the feasible region determined by the system of linear constraints are $(0,3), (1,1)$ and $(3,0) .$ Let $Z=p x+q y,$ where $p, q\,>\,0 .$ Condition on $p$ and $q,$ so that the maximum of $Z$ occurs at $(3,0)$ and $(1,1)$ is $.....$

The sum of $1 + n\left( {1 - \frac{1}{x}} \right) + \frac{{n(n + 1)}}{{2!}}{\rm{ }}{\left( {1 - \frac{1}{x}} \right)^2} + .....\infty ,$ will be

If $f'(x) = {x^2} + 5$ and $f(0) = - 1$, then $f(x) = $

If $\frac{{1 - {{\tan }^2}\theta }}{{{{\sec }^2}\theta }} = \frac{1}{2}$, then the general value of $\theta $ is

If $f (x) = \frac{{{{\log }_{\sin |x|}}{{\cos }^3}x}}{{{{\log }_{\sin |3x|}}{{\cos }^3}\left( {\frac{x}{2}} \right)}}for |x| <\frac{\pi }{3} x \ne 0= 4$ for $x = 0$then, the number of points of discontinuity of f in $\left( { - \frac{\pi }{3},\,\frac{\pi }{3}} \right)$ is

Number of points where the function $f(x) = $ maximum $\left( {\sqrt {2x - {x^2}} ,2 - x} \right)$ is non differentiable is