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 determine the intervals of increase and decrease by testing the sign of the derivative at points in each interval determined by the critical points.

Choose a test point to the left of -√6, between -√6 and √6, and to the right of √6.

For x < -√6, choose x = -√7. Substituting into the derivative gives a positive result, so the function is increasing on the interval (-∞, -√6).

For -√6 < x < √6, choose x = 0. Substituting into the derivative gives a positive result, so the function is increasing on the interval (-√6, √6).

For x > √6, choose x = √7. Substituting into the derivative gives a negative result, so the function is decreasing on the interval (√6, ∞).

Step 4: Find the x-values of all relative extrema The x-values of all relative extrema are the x-values where the function changes from increasing to decreasing or vice versa.

From our intervals of increase and decrease, we see that the function changes from increasing to decreasing at x = √6, so there is a relative maximum at x = √6.

Similarly, the function changes from decreasing to increasing at x = -√6, so there is a relative minimum at x = -√6.