1. **State the problem:** We are given the function $f(x) = x^3 - 6x^2 + 9x + 2$ and asked to find intervals where $f$ is increasing or decreasing using the graph and then verify by finding $f'(x)$ and its sign diagram. 2. **Find the derivative:** The derivative of $f(x)$ is given by the power rule: $$f'(x) = 3x^2 - 12x + 9$$ 3. **Factorize the derivative:** $$f'(x) = 3(x^2 - 4x + 3) = 3(x - 1)(x - 3)$$ 4. **Find critical points:** Set $f'(x) = 0$: $$3(x - 1)(x - 3) = 0 \implies x = 1 \text{ or } x = 3$$ 5. **Construct sign diagram for $f'(x)$:** - For $x < 1$, choose $x=0$: $f'(0) = 3(0-1)(0-3) = 3(-1)(-3) = 9 > 0$ - For $1 < x < 3$, choose $x=2$: $f'(2) = 3(2-1)(2-3) = 3(1)(-1) = -3 < 0$ - For $x > 3$, choose $x=4$: $f'(4) = 3(4-1)(4-3) = 3(3)(1) = 9 > 0$ 6. **Interpret sign diagram:** - $f'(x) > 0$ means $f$ is increasing. - $f'(x) < 0$ means $f$ is decreasing. 7. **Answer intervals:** - Increasing on $(-\infty, 1)$ and $(3, \infty)$ - Decreasing on $(1, 3)$ 8. **Check with graph:** The graph shows the function increasing before $x=1$, decreasing between $x=1$ and $x=3$, and increasing again after $x=3$, confirming our derivative analysis. Final answer: - Increasing intervals: $(-\infty, 1) \cup (3, \infty)$ - Decreasing interval: $(1, 3)$