3.3 Enhanced Homework Part 1 of 10 Points: 0 of 1 concave down, where any points of inflection occur, and where any intercepts occur. \[ f(x)=\frac{-3}{x-7} \] On what interval(s) is $f$ increasing and on what interval(s) is $f$ decreasing? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function is decreasing on $\square$. The function is never increasing. (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.) B. The function is increasing on $\square$ and decreasing on $\square$. (Simplify your answers. Type your answers in interval notation. Type exact answers, using radicals as needed. Use a comma to separate answers as needed.) C. The function is increasing on $\square$. The function is never decreasing. (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.) D. The function is never increasing or decreasing.
Solution
Step 1 ：The function given is \(f(x)=\frac{-3}{x-7}\).
Step 2 ：To find where the function is increasing or decreasing, we need to find its derivative and analyze its sign.
Step 3 ：The derivative of a function gives us the rate of change of the function. If the derivative is positive, the function is increasing. If the derivative is negative, the function is decreasing.
Step 4 ：The derivative of the function is \(f'(x)=\frac{3}{{(x-7)}^2}\), which is always positive for all \(x \neq 7\).
Step 5 ：Therefore, the function is increasing on the intervals \((-\infty, 7)\) and \((7, \infty)\).
Step 6 ：The function is never decreasing.
Step 7 ：\(\boxed{\text{Final Answer: The function is increasing on } (-\infty, 7) \text{ and } (7, \infty). \text{ The function is never decreasing.}}\)
