Part 2 of 5
For the following function, a) give the coordinates of any critical points and classify each point as a relative maximum, a relative minimum, or neither; b) identify intervals where the function is increasing or decreasing; $c$ ) give the coordinates of any points of inflection; d) identify intervals where the function is concave up or concave down, and e) sketch the graph.
$h (x) = 3 x^{3} - 9 x$
b) On what interval(s) is $h$ increasing or decreasing? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
A. The function is increasing on $◻$ . The function is decreasing on (Simplify your answers. Type your answers in interval notation. Use a comma to separate answers as needed.)
B. The function is increasing on The function is never decreasing.
(Simnlify your answer. Type your answer in interval notation. Use a comma to separate answers

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Answer

$The function is increasing on (- \infty, - 1) and (1, \infty) . The function is decreasing on (- 1, 1) .$

Steps

Step 1 ：Find the derivative of the function $h (x) = 3 x^{3} - 9 x$ .

Step 2 ：The derivative of the function is $h^{'} (x) = 9 x^{2} - 9$ .

Step 3 ：Set the derivative equal to zero and solve for x to find the critical points: $9 x^{2} - 9 = 0$ .

Step 4 ：The critical points are -1 and 1.

Step 5 ：Test the intervals around the critical points by plugging in values into the derivative. We will test the intervals $(- \infty, - 1)$ , $(- 1, 1)$ , and $(1, \infty)$ .

Step 6 ：The function is increasing on the intervals $(- \infty, - 1)$ and $(1, \infty)$ , and decreasing on the interval $(- 1, 1)$ .

Step 7 ： $The function is increasing on (- \infty, - 1) and (1, \infty) . The function is decreasing on (- 1, 1) .$

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