Let = _ k =1 ^ ^ 2 k ( 6 ) Let g:[0,1] R be the function defined … - Vidyadip

MCQ

Let

$\alpha=\sum_{ k =1}^{\infty} \sin ^{2 k}\left(\frac{\pi}{6}\right)$

Let $g:[0,1] \rightarrow R$ be the function defined by

$g( x )=2^{\alpha x }+2^{\alpha(1- x )}$

Then, which of the following statements is/are $TRUE$?

$(A)$ The minimum value of $g( x )$ is $2^{\frac{7}{6}}$

$(B)$ The maximum value of $g( x )$ is $1+2^{\frac{1}{3}}$

$(C)$ The function $g( x )$ attains its maximum at more than one point

$(D)$ The function $g( x )$ attains its minimum at more than one point

A
$A,B$
B
$A,B,D$
C
$A,C$
✓
$A,B,C$

✓

Answer

Correct option: D.

$A,B,C$

d
$\alpha=\left(\frac{1}{2}\right)^2+\left(\frac{1}{2}\right)^4+\left(\frac{1}{2}\right)^6+\ldots$

$\alpha=\frac{\frac{1}{4}}{1-\frac{1}{4}}=\frac{1}{3}$

$\therefore g(x)=2^{x / 3}+2^{1 / 3(1-x)}$

$\therefore g(x)=2^{x / 3}+\frac{2^{1 / 3}}{2^{x / 3}}$

$\text { where } g(0)=1+2^{1 / 3} \& g(1)=1+2^{1 / 3}$

$\therefore g^{\prime}(x)=\frac{1}{3}\left(2^{x / 3}-\frac{2^{1 / 3}}{2^{\pi / 3}}\right)=0$

$\Rightarrow 2^{2 x / 3}=2^{1 / 3} \Rightarrow x=\frac{1}{2}=\text { critical point }$

(image)

& $g\left(\frac{1}{2}\right)=2^{\frac{7}{6}}$

$\therefore \text { graph of } g(x) \text { in }[0,1]$

(image)

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