1. **Problem Statement:** Analyze the function $$f(x) = x^3 + 6x^2 + 9x + 2$$ for its relative extrema, intervals of increase/decrease, concavity, and points of inflection. 2. **Find the first derivative:** $$f'(x) = \frac{d}{dx}(x^3 + 6x^2 + 9x + 2) = 3x^2 + 12x + 9$$ 3. **Find critical points by setting $$f'(x) = 0$$:** $$3x^2 + 12x + 9 = 0$$ Divide both sides by 3: $$x^2 + 4x + 3 = 0$$ Factor: $$(x + 3)(x + 1) = 0$$ So, critical points are $$x = -3$$ and $$x = -1$$. 4. **Determine intervals of increase/decrease:** - For $$x < -3$$, choose $$x = -4$$: $$f'(-4) = 3(-4)^2 + 12(-4) + 9 = 48 - 48 + 9 = 9 > 0$$, so $$f$$ is increasing. - For $$-3 < x < -1$$, choose $$x = -2$$: $$f'(-2) = 3(4) + 12(-2) + 9 = 12 - 24 + 9 = -3 < 0$$, so $$f$$ is decreasing. - For $$x > -1$$, choose $$x = 0$$: $$f'(0) = 9 > 0$$, so $$f$$ is increasing. 5. **Classify critical points using the first derivative test:** - At $$x = -3$$, $$f'$$ changes from positive to negative, so $$f$$ has a relative maximum. - At $$x = -1$$, $$f'$$ changes from negative to positive, so $$f$$ has a relative minimum. 6. **Find the second derivative:** $$f''(x) = \frac{d}{dx}(3x^2 + 12x + 9) = 6x + 12$$ 7. **Find points of inflection by setting $$f''(x) = 0$$:** $$6x + 12 = 0 \Rightarrow x = -2$$ 8. **Determine concavity intervals:** - For $$x < -2$$, choose $$x = -3$$: $$f''(-3) = 6(-3) + 12 = -18 + 12 = -6 < 0$$, so $$f$$ is concave downward. - For $$x > -2$$, choose $$x = 0$$: $$f''(0) = 12 > 0$$, so $$f$$ is concave upward. 9. **Summary:** - Relative maximum at $$x = -3$$. - Relative minimum at $$x = -1$$. - Increasing on $$(-\infty, -3) \cup (-1, \infty)$$. - Decreasing on $$(-3, -1)$$. - Concave downward on $$(-\infty, -2)$$. - Concave upward on $$(-2, \infty)$$. - Point of inflection at $$x = -2$$. 10. **Function values at key points:** - $$f(-3) = (-3)^3 + 6(-3)^2 + 9(-3) + 2 = -27 + 54 - 27 + 2 = 2$$ - $$f(-1) = (-1)^3 + 6(-1)^2 + 9(-1) + 2 = -1 + 6 - 9 + 2 = -2$$ - $$f(-2) = (-2)^3 + 6(-2)^2 + 9(-2) + 2 = -8 + 24 - 18 + 2 = 0$$ These points help sketch the graph showing the shape and behavior of $$f(x)$$.