Let $\alpha=\sum_{k=1}^{\infty} \sin ^{2 k}\left(\frac{\pi}{6}\right)$. Let $g:[0,1] \rightarrow \mathbb{R}$ be the function defined by $g(x)=2^{\alpha x}+2^{\alpha(1-x)}$. Then, which of the following statements is/are TRUE ?

Let α=∑ₖ₌₁^∞ sin ^(2 k)(π/6). Let g:[0,1] → R be the function defined by g(x)=2^(α x)+2^(α(1-x)). Then, which of the following statements is/are TRUE ? $\alpha = \sum_{k=1}^{\infty} \sin^{2k}\!\left(\dfrac{\pi}{6}\right)$ $g(x) = 2^{\alpha x} + 2^{\alpha(1-x)}, \qquad x \in [0,1]$ $\text{Check: the min and max of } g \text{ on } [0,1],$ $\text{and whether each is attained at more than one point}$ $\sin\dfrac{\pi}{6} = \dfrac{1}{2}$ $\alpha = \sum_{k=1}^{\infty}\left(\dfrac{1}{4}\right)^{k} = \dfrac{1/4}{1 - 1/4} = \dfrac{1}{3}$ $g(x) = 2^{x/3} + 2^{(1-x)/3}, \qquad x \in [0,1]$ $2^{x/3}\cdot 2^{(1-x)/3} = 2^{1/3} \quad (\text{constant})$ $g(x) \ \ge \ 2\sqrt{2^{1/3}} = 2^{7/6}$ $x/3 = (1-x)/3 \ \Rightarrow \ x = \dfrac{1}{2}$ $g_{\min} = 2^{7/6} \text{ at the single point } x = \dfrac{1}{2}$ $u = 2^{x/3} \in [1,\ 2^{1/3}], \qquad g = u + \dfrac{2^{1/3}}{u}$ $g \text{ is convex in } u \Rightarrow \text{the max is at an endpoint}$ $g(0) = 1 + 2^{1/3} = g(1)$ $g_{\max} = 1 + 2^{1/3} \text{ at } x=0 \text{ and } x=1$ $(1)\ \checkmark \quad (2)\ \checkmark \quad (3)\ \checkmark \quad (4)\ \times$ $\boxed{\text{Answer: (1), (2), (3)}}$