sketch a graph of the function. \[ F(x)=2-2 x^{3} \] Describe any relative extrema. Select the correct choice below and, if necessary, fill in the answer box(es) to within your choice. A. The relative minimum point(s) is/are and the relative maximum point(s) is/are (Simplify your answers. Type ordered pairs, using integers or fractions. Use a comma to separate answers as needed.) B. The relative minimum point(s) is/are and there are no relative maximum points. (Simplify your answer. Type an ordered pair, using integers or fractions. Use a comma to separate answers as needed.) C. The relative maximum point(s) is/are and there are no relative minimum points. (Simplify your answer. Type an ordered pair, using integers or fractions. Use a comma to separate answers as needed.) D. There are no relative minimum points and there are no relative maximum points. Identify any intervals over which the function is increasing or decreasing. Select the correct choice below and fill in the answer box(es) within your choice. A. The function $F(x)$ is increasing over the interval(s) $\square$ and is not decreasing anywhere. (Type your answer in interval notation. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) B. The function $F(x)$ is decreasing over the interval(s) $\square$ and is not increasing anywhere. (Type your answer in interval notation. Use integers or fractions for âny numbers in the expression. Use a comma to separate answers as needed.) C. The function $F(x)$ is increasing over the interval(s) $\square$ and decreasing over the interval(s) (Type your answers in interval notation. Use integers or fractions for any numbers in the expressions. Use a comma to separate answers as needed.)
Solution
Step 1 ：Given the function \(F(x)=2-2 x^{3}\), we first find the derivative of the function, which is \(F'(x)=-6x^{2}\).
Step 2 ：We then set the derivative equal to zero to find the critical points: \(-6x^{2}=0\), which gives us \(x=0\).
Step 3 ：We then use the second derivative test to determine whether these points are relative minima, maxima, or neither. The second derivative of the function is \(F''(x)=-12x\). At \(x=0\), the second derivative is also 0, which means it's not a relative extrema.
Step 4 ：Therefore, there are no relative minimum points and there are no relative maximum points.
Step 5 ：To find the intervals over which the function is increasing or decreasing, we again use the derivative. The function is increasing where the derivative is positive and decreasing where the derivative is negative. In this case, the derivative is negative for all \(x\) except 0, so the function is decreasing over the entire real line except at \(x=0\).
Step 6 ：Final Answer: \(\boxed{\text{The correct choice for the relative extrema is D. There are no relative minimum points and there are no relative maximum points.}}\)
Step 7 ：Final Answer: \(\boxed{\text{The correct choice for the intervals of increase and decrease is B. The function } F(x) \text{ is decreasing over the interval(s) } (-\infty, 0) \cup (0, \infty) \text{ and is not increasing anywhere.}}\)
