Question

Let
$\alpha =\sum _{k=1}^{\infty }{\sin }^{2k}\left(\frac{\pi }{6}\right)$
Let $g:[0,1]\to R$ 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

Answer 1

Answer: (A), (B), (C)

$\alpha ={\left(\frac{1}{2}\right)}^2+{\left(\frac{1}{2}\right)}^4+{\left(\frac{1}{2}\right)}^6+\dots$
$\alpha =\frac{\frac{1}{4}}{1-\frac{1}{4}}=\frac{1}{3}$
$\therefore g(x)=2^{x/3}+2^{1/3(1-x)}$
$\therefore g(x)=2^{x/3}+\frac{2^{1/3}}{2^{x/3}}$
where $g(0)=1+2^{1/3}\&g(1)=1+2^{1/3}$
$\therefore g^{\prime }(x)=\frac{1}{3}\left(2^{x/3}-\frac{2^{1/3}}{2^{x/3}}\right)=0$
$\Rightarrow 2^{2x/3}=2^{1/3}\Rightarrow x=\frac{1}{2}=$ critical point