Problem
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 $\mathrm{h}$ 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 $\square$. 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
Solution
Step 1 ：Find the derivative of the function \(h(x)=3x^3-9x\).
Step 2 ：The derivative of the function is \(h'(x)=9x^2-9\).
Step 3 ：Set the derivative equal to zero and solve for x to find the critical points: \(9x^2-9=0\).
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 ：\(\boxed{\text{The function is increasing on } (-\infty, -1) \text{ and } (1, \infty). \text{ The function is decreasing on } (-1, 1).}\)
