Question

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

Answer 1

Answer: (C)

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

Answer 2

We have,
$\alpha=\sum\limits^{∞}_{k=1}(\frac{1}{2})^{2k}=\frac{\frac{1}{4}}{1-\frac{1}{4}}=\frac{1}{3}$
$g(x)=2^{\frac{7}{3}}+2^{\frac{1-x}{3}}$
$\frac{2^{\frac{x}{3}}+2^{\frac{1-x}{3}}}{2}≥\left(2^{\frac{x}{3}+\frac{1-x}{3}}\right)^{\frac{1}{2}}$
$⇒g(x)≥2^{\frac{7}{6}}$
Also g(x) ≤ 1 + 21/3 at x = 0, 1
So, the correct options are as follows :
(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