1. The problem is to determine the intervals where a function is increasing or decreasing. 2. To find these intervals, we use the first derivative test. The function is increasing where its derivative $f'(x) > 0$ and decreasing where $f'(x) < 0$. 3. Steps: - Find the derivative $f'(x)$ of the function $f(x)$. - Solve $f'(x) = 0$ to find critical points. - Use these critical points to divide the domain into intervals. - Test the sign of $f'(x)$ in each interval. 4. For example, if $f(x) = x^3 - 3x^2 + 4$, then: - $f'(x) = 3x^2 - 6x$ - Solve $3x^2 - 6x = 0$ which factors as $3x(x - 2) = 0$ giving critical points $x=0$ and $x=2$. 5. Test intervals: - For $x < 0$, pick $x = -1$: $f'(-1) = 3(-1)^2 - 6(-1) = 3 + 6 = 9 > 0$ so increasing. - For $0 < x < 2$, pick $x=1$: $f'(1) = 3(1)^2 - 6(1) = 3 - 6 = -3 < 0$ so decreasing. - For $x > 2$, pick $x=3$: $f'(3) = 3(3)^2 - 6(3) = 27 - 18 = 9 > 0$ so increasing. 6. Therefore, the function is increasing on $(-\infty, 0)$ and $(2, \infty)$, and decreasing on $(0, 2)$. This method applies to any differentiable function to find intervals of increase and decrease.