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)=3⋅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)=3⋅𝑥−𝑥³ is 𝑓'(𝑥)=3-3𝑥².

Step 2: Set the derivative equal to zero and solve for x Setting the derivative equal to zero gives us 3-3𝑥²=0. Solving for x gives us x=±1.

Step 3: Determine the intervals of increase and decrease To determine where the function is increasing or decreasing, we test the intervals determined by the critical numbers (x=±1) in the derivative.

For x<-1, 𝑓'(𝑥)=3-3𝑥² is positive, so the function is increasing on the interval (-∞, -1).

For -1<x<1, 𝑓'(𝑥)=3-3𝑥² is negative, so the function is decreasing on the interval (-1, 1).

For x>1, 𝑓'(𝑥)=3-3𝑥² is positive, so the function is increasing on the interval (1, ∞).

Step 4: Find the relative extrema The function has a relative maximum at x=-1 and a relative minimum at x=1. This is because the function changes from increasing to decreasing at x=-1 (making it a relative maximum), and from decreasing to increasing at x=1 (making it a relative minimum).