At which \( x \)-intercept(s) does the graph touch the \( x \)-axis and turn around? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( x=\square \) (Use a comma to separate answers as needed.). B. There are no \( x \)-intercepts at which the graph touches the \( x \)-axis and turns around.

Eveluate \( \frac{\ln \sqrt{60}-\ln \sqrt{20}-\ln \sqrt{2}}{\ln 9-\ln 1} \)

Q. 41. Find all asymptotes of the curve \( x^{3}+y^{3}=3 a x y \). (Bhopal 1991, 2000; Sagar

4. Найдите область определения функции \( y=1-\sqrt{3 t g \frac{x}{2}-\sqrt{3}} \).

I. Work Out (Show the necessary steps) For the rational functions (a-d) given below answer question (i-vi) each of the following i. Find the domain ii. \( \quad \) Find hole, if any iii. Find the vertical asymptote(s), if any iv. Find the horizontal or oblique asymptote, if any v. Find the \( x \)-intercept(s) and \( y \) - intercept(s) vi. \( \quad \) Prepare sign chart vii. Sketch the graph of each rational function a. \( \quad f(x)=\frac{x+2}{x^{2}-9} \)

The population of a certain species in a protected area can be modeled by the function \( P(t) = 200 + 50 \sin(t) \), where \( t \) is measured in years. Determine the total population increase over one complete cycle (from \( t=0 \) to \( t=2\pi \)).