Which of the following statements is correct?

Let $f(x) = \frac{2^{x+2} + 16}{2^{2x+1} + 2^{x+4} + 32}$. Then the value of $8 \left( f \left( \frac{1}{15} \right) + f \left( \frac{2}{15} \right) + \dots + f \left( \frac{59}{15} \right) \right)$ is equal to

Let $f(x)$ and $g(x)$ be two continuous functions defined from $R \rightarrow R$,such that $f(x_1) > f(x_2)$ and $g(x_1) < g(x_2)$ for all $x_1 > x_2$. Then the solution set of $f(g(\alpha^2 - 2\alpha)) > f(g(3\alpha - 4))$ is

If $a+\alpha=1, b+\beta=2$ and $af(x)+\alpha f\left(\frac{1}{x}\right)=bx+\frac{\beta}{x}$ for $x \neq 0$,then the value of the expression $\frac{f(x)+f\left(\frac{1}{x}\right)}{x+\frac{1}{x}}$ is ..... .

Which one of the following is not bounded on the intervals as indicated?

Let $R$ denote the set of all real numbers. Let $f: R \rightarrow R$ be a function such that $f(x) > 0$ for all $x \in R$,and $f(x+y)=f(x) f(y)$ for all $x, y \in R$. Let the real numbers $a_1, a_2, \ldots, a_{50}$ be in an arithmetic progression. If $f(a_{31})=64 f(a_{25})$,and $\sum_{i=1}^{50} f(a_i)=3(2^{25}+1)$,then the value of $\sum_{i=6}^{30} f(a_i)$ is: