We are given the cubic polynomial , and we are asked to find the interval where the function is increasing.
1. Step 1: Find the derivative of the function. The first derivative of will tell us where the function is increasing or decreasing: Using standard differentiation rules:
2. Step 2: Find the critical points by setting the derivative equal to zero. Set to find the critical points: Divide through by 6: Factor the quadratic equation: The critical points are and .
3. Step 3: Test the intervals around the critical points. We now test the sign of in the intervals , , and . - For (in ), substitute into : - For (in ), substitute into : - For (in ), substitute into :
4. Step 4: Determine the intervals of increase and decrease. - The function is increasing where , which occurs in the intervals and . - The function is decreasing in the interval . Thus, the function is increasing in the interval .