1. **Problem statement:** Given the function $$g(x) = x^3 - 3x^2 - 9x - 5,$$ we will find its derivatives, stationary points, nature of stationary points, point of inflection, and sketch the graph labeling key points. 2. **Find the first and second derivatives:** $$g'(x) = \frac{d}{dx}(x^3 - 3x^2 - 9x - 5) = 3x^2 - 6x - 9$$ $$g''(x) = \frac{d}{dx}(3x^2 - 6x - 9) = 6x - 6$$ 3. **Find stationary points:** Stationary points occur where $$g'(x) = 0$$ Solve: $$3x^2 - 6x - 9 = 0$$ Divide both sides by 3: $$x^2 - 2x - 3 = 0$$ Factor: $$(x - 3)(x + 1) = 0$$ So stationary points at $$x = 3$$ and $$x = -1$$ 4. **Find coordinates of stationary points:** Calculate $$g(x)$$ at $$x = -1$$: $$g(-1) = (-1)^3 - 3(-1)^2 - 9(-1) - 5 = -1 - 3 + 9 - 5 = 0$$ So stationary point at $$(-1, 0)$$ Calculate $$g(3)$$: $$g(3) = 3^3 - 3(3)^2 - 9(3) - 5 = 27 - 27 - 27 - 5 = -32$$ So stationary point at $$(3, -32)$$ 5. **Determine the nature of stationary points:** Evaluate $$g''(x)$$ at each stationary point: At $$x = -1$$: $$g''(-1) = 6(-1) - 6 = -6 - 6 = -12 < 0$$ so $$(-1,0)$$ is a local maximum. At $$x = 3$$: $$g''(3) = 6(3) - 6 = 18 - 6 = 12 > 0$$ so $$(3, -32)$$ is a local minimum. 6. **Find the inflection point:** Inflection points occur where $$g''(x) = 0$$ Solve: $$6x - 6 = 0$$ $$6x = 6$$ $$x = 1$$ Calculate $$g(1)$$: $$g(1) = 1^3 - 3(1)^2 - 9(1) - 5 = 1 - 3 - 9 - 5 = -16$$ Inflection point at $$(1, -16)$$ 7. **Summary of key points:** - Stationary points: $$(-1, 0)$$ (local max), $$(3, -32)$$ (local min) - Point of inflection: $$(1, -16)$$ - x-intercepts given at $$(-1, 0)$$ and $$(5, 0)$$ 8. **Graph sketch instructions:** Plot $g(x)$ on Cartesian plane, marking: - Stationary points: $$(-1,0)$$ and $$(3,-32)$$ - Inflection point: $$(1, -16)$$ - x-intercepts: $$(-1,0)$$ and $$(5,0)$$ Graph rises to local max at $$(-1, 0)$$, dips to local min at $$(3, -32)$$, and passes through the inflection point $$(1, -16)$$ where concavity changes. Final answer: Stationary points: $$(-1,0), (3,-32)$$ Nature: $$(-1,0)$$ local max, $$(3,-32)$$ local min Inflection point: $$(1, -16)$$