Find the open intervals on which the function 𝑓 is increasing or decreasing, and find the 𝑥-values of all relative extrema (turning points).𝑓(𝑥)=18⋅𝑥−𝑥3

To find the open intervals on which the function 𝑓 is increasing or decreasing, and the 𝑥-values of all relative extrema (turning points), we need to follow these steps:

Step 1: Find the derivative of the function The derivative of the function 𝑓(𝑥)=18⋅𝑥−𝑥³ is 𝑓'(𝑥)=18-3𝑥².

Step 2: Set the derivative equal to zero and solve for x Setting the derivative equal to zero gives us 18-3𝑥²=0. Solving for x gives us x=±√6.

Step 3: Determine the intervals of increase and decrease We can use a number line to determine where the function is increasing and decreasing. We test the intervals (-∞, -√6), (-√6, √6), and (√6, ∞) by choosing test points in these intervals and substituting them into the derivative.

For the interval (-∞, -√6), we can choose x=-√7. Substituting this into the derivative gives us a positive value, so the function is increasing on this interval.

For the interval (-√6, √6), we can choose x=0. Substituting this into the derivative gives us a positive value, so the function is increasing on this interval.

For the interval (√6, ∞), we can choose x=√7. Substituting this into the derivative gives us a negative value, so the function is decreasing on this interval.

Step 4: Find the x-values of all relative extrema The x-values of all relative extrema are the values of x where the function changes from increasing to decreasing or vice versa. From our number line, we can see that the function changes from increasing to decreasing at x=√6, so this is a relative maximum. The function does not change from decreasing to increasing, so there are no relative minima.

So, the function is increasing on the intervals (-∞, -√6) and (-√6, √6), and decreasing on the interval (√6, ∞). The x-value of the relative maximum is √6.