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. \[ h(x)=5 x^{3}-15 x \] a) What are the coordinates of the relative extrema? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The relative minimum point(s) is/are $\square$ and the relative maximum point(s) is/are (Simplify your answers. Use integers or fractions for any numbers in the expression. Type an ordered pair. Use a comma to separate answers as needed.) B. The relative minimum point(s) is/are $\square$ 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 $\square$ 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.
Solution
Step 1 ：The function given is \(h(x) = 5x^3 - 15x\).
Step 2 ：To find the relative extrema, we need to find the critical points of the function. Critical points occur where the derivative of the function is zero or undefined.
Step 3 ：The derivative of the function \(h(x) = 5x^3 - 15x\) is \(h'(x) = 15x^2 - 15\).
Step 4 ：Setting this equal to zero gives us the critical points \(x = -1\) and \(x = 1\).
Step 5 ：We can determine whether each point is a relative maximum, a relative minimum, or neither by using the second derivative test. The second derivative of the function is \(h''(x) = 30x\).
Step 6 ：If the second derivative at a critical point is positive, the point is a relative minimum. If it's negative, the point is a relative maximum. If it's zero, the test is inconclusive.
Step 7 ：Applying the second derivative test, we find that the point \((-1, 10)\) is a relative maximum and the point \((1, -10)\) is a relative minimum.
Step 8 ：\(\boxed{\text{Final Answer: The relative minimum point is (1, -10) and the relative maximum point is (-1, 10).}}\)
