(1 point) Let $F(x)=\int_{2}^{x} \frac{8}{\ln (5 t)} d t$, for $x \geq 2$
A. $F^{\prime}(x)=$
B. On what interval or intervals is $F$ increasing?
$x \in$
(Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or $(1,5),(7,10)$, or enter none for no intervals.)
C. On what interval or intervals is the graph of $F$ concave up?
$x \in$
(Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or $(1,5),(7,10)$, or enter none for no intervals.)
Use your answers above and your knowledge of derivatives and antiderivatives to sketch a graph of $F(x)$.

Step 1 ：First, we need to find the derivative of the function $F(x)$, which is given by the Fundamental Theorem of Calculus as $F^{\prime}(x) = \frac{8}{\ln(5x)}$.

Step 2 ：Next, we need to find the intervals where $F$ is increasing. This happens when $F^{\prime}(x) > 0$. Since the denominator of $F^{\prime}(x)$ is always positive for $x > 2$, we only need to consider the numerator. The function $F$ is increasing for all $x > 2$.

Step 3 ：Then, we need to find the intervals where the graph of $F$ is concave up. This happens when the second derivative of $F$, $F^{\prime\prime}(x)$, is greater than 0. The second derivative is $F^{\prime\prime}(x) = -\frac{8}{x(\ln(5x))^2}$.

Step 4 ：Since the numerator of $F^{\prime\prime}(x)$ is negative, the function $F$ is concave up when the denominator is negative. However, the denominator is always positive for $x > 2$. Therefore, the graph of $F$ is never concave up.

Step 5 ：Finally, we can sketch the graph of $F(x)$ using the information we have found. The function $F$ is increasing for all $x > 2$ and is never concave up. Therefore, the graph of $F$ is a curve that starts at some point on the y-axis and increases without bound as $x$ increases, but never curves upwards.

Step 6 ：Thus, the answer is $F^{\prime}(x) = \boxed{\frac{8}{\ln(5x)}}$, $F$ is increasing on the interval $x \in \boxed{(2,\infty)}$, and the graph of $F$ is never concave up.