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

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

Answer 2

The correct answers are:
(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