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. \[ h(x)=5 x^{3}-15 x \] B. The relative minimum point(s) is/are and there are no relative maximum point(s). (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an ordered pair. Use a comma to separate answers as needed.) C. The relative maximum point(s) is/are and there are no relative minimum point(s). (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an ordered pair. Use a comma to separate answers as needed.) D. There are no relative minimum points and there are no relative maximum points. 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 $(-\infty,-1),(1, \infty)$. The function is decreasing on $(-1,1)$. (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. (Simplify your answer. Type your answer in interval notation. Use a comma to separate answers as needed.) C. The function is decreasing on . The function is never increasing. (Simplify your answer. Type your answer in interval notation. Use a comma to separate answers as needed.) D. The function is never increasing or decreasing. c) What are the coordinates of the inflection point(s)? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The coordinates of the inflection point(s) are (Simplify your answer. Type an ordered pair. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) B. There are no inflection points.
Solution
Step 1 ：Find the derivative of the function \(h(x) = 5x^3 - 15x\), which is \(h'(x) = 15x^2 - 15\).
Step 2 ：Set the derivative equal to zero to find the critical points: \(15x^2 - 15 = 0\). Solving this equation gives the critical points \(x = -1\) and \(x = 1\).
Step 3 ：Classify the critical points as relative maximum, minimum, or neither by checking the sign of the derivative on either side of these points. The point \((-1, -20)\) is a relative maximum and the point \((1, 20)\) is a relative minimum.
Step 4 ：Find the second derivative of the function, which is \(h''(x) = 30x\).
Step 5 ：Set the second derivative equal to zero to find potential points of inflection: \(30x = 0\). Solving this equation gives the potential point of inflection \(x = 0\).
Step 6 ：Check the sign of the second derivative on either side of the potential point of inflection to confirm if it is a point of inflection. The point \((0, 0)\) is a point of inflection.
Step 7 ：Determine the intervals where the function is increasing or decreasing. The function is increasing on the intervals \((-\infty, -1)\) and \((1, \infty)\), and decreasing on the interval \((-1, 1)\).
Step 8 ：Determine the intervals where the function is concave up or concave down. The function is concave up on the interval \((0, \infty)\) and concave down on the interval \((-\infty, 0)\).
Step 9 ：\(\boxed{\text{Final Answer: The relative maximum point is } (-1, -20) \text{ and the relative minimum point is } (1, 20). \text{ The function is increasing on the intervals } (-\infty, -1) \text{ and } (1, \infty), \text{ and decreasing on the interval } (-1, 1). \text{ The point of inflection is at } (0, 0). \text{ The function is concave up on the interval } (0, \infty) \text{ and concave down on the interval } (-\infty, 0).}\)
