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 $◻$ . 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 $◻$ and decreasing on $◻$ .
(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 $◻$ . 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.

Answer

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Answer

$Final Answer: The function is increasing on (- \infty, 7) and (7, \infty) . The function is never decreasing.$

Steps

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 ： $Final Answer: The function is increasing on (- \infty, 7) and (7, \infty) . The function is never decreasing.$

Answer

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