1. **Problem Statement:** We are asked to analyze the function $$F(x) = x^4 - 4x^3 + 10$$ by finding its extrema, intervals of increase/decrease, concavity, and then sketching its graph with key points. 2. **Step 1: Find the first derivative to locate extrema.** The first derivative is $$F'(x) = \frac{d}{dx}(x^4 - 4x^3 + 10) = 4x^3 - 12x^2.$$ 3. **Step 2: Find critical points by setting $$F'(x) = 0$$.** $$4x^3 - 12x^2 = 0 \implies 4x^2(x - 3) = 0.$$ So, critical points are at $$x = 0$$ and $$x = 3$$. 4. **Step 3: Determine intervals of increase and decrease using the first derivative test.** - For $$x < 0$$, pick $$x = -1$$: $$F'(-1) = 4(-1)^3 - 12(-1)^2 = -4 - 12 = -16 < 0$$, so decreasing. - For $$0 < x < 3$$, pick $$x = 1$$: $$F'(1) = 4(1)^3 - 12(1)^2 = 4 - 12 = -8 < 0$$, still decreasing. - For $$x > 3$$, pick $$x = 4$$: $$F'(4) = 4(64) - 12(16) = 256 - 192 = 64 > 0$$, increasing. Thus, $$F$$ is decreasing on $$(-\infty, 3)$$ and increasing on $$(3, \infty)$$. 5. **Step 4: Find the second derivative to analyze concavity.** $$F''(x) = \frac{d}{dx}(4x^3 - 12x^2) = 12x^2 - 24x.$$ 6. **Step 5: Find inflection points by setting $$F''(x) = 0$$.** $$12x^2 - 24x = 0 \implies 12x(x - 2) = 0.$$ So, inflection points at $$x = 0$$ and $$x = 2$$. 7. **Step 6: Determine concavity intervals using the second derivative test.** - For $$x < 0$$, pick $$x = -1$$: $$F''(-1) = 12(1) + 24 = 36 > 0$$, concave up. - For $$0 < x < 2$$, pick $$x = 1$$: $$F''(1) = 12 - 24 = -12 < 0$$, concave down. - For $$x > 2$$, pick $$x = 3$$: $$F''(3) = 108 - 72 = 36 > 0$$, concave up. 8. **Step 7: Calculate function values at critical and inflection points for plotting.** - $$F(0) = 0 - 0 + 10 = 10$$ (local maximum candidate) - $$F(3) = 81 - 108 + 10 = -17$$ (local minimum candidate) - $$F(2) = 16 - 32 + 10 = -6$$ (inflection point) 9. **Step 8: Summary of key points and behavior:** - Local maximum at $$(0, 10)$$ - Local minimum at $$(3, -17)$$ - Inflection points at $$(0, 10)$$ and $$(2, -6)$$ - Decreasing on $$(-\infty, 3)$$, increasing on $$(3, \infty)$$ - Concave up on $$(-\infty, 0) \cup (2, \infty)$$, concave down on $$(0, 2)$$ 10. **Step 9: Sketch the graph** The graph starts concave up and decreasing from the left, reaches a local maximum at $x=0$, then decreases concave down until $x=2$ (inflection), continues decreasing to local minimum at $x=3$, then increases concave up thereafter. Final answer: $$\boxed{\text{Local max at } (0,10), \text{ local min at } (3,-17), \text{ inflection points at } (0,10) \text{ and } (2,-6)}$$