1. **Problem Statement:** Find the critical points, concavity, and points of inflection for the function $$y = x \sqrt{4 - x^2}$$ and graph it. 2. **Step 1: Rewrite the function for differentiation.** We have $$y = x (4 - x^2)^{1/2}$$. 3. **Step 2: Find the first derivative using the product rule:** $$y' = (4 - x^2)^{1/2} + x \cdot \frac{1}{2}(4 - x^2)^{-1/2}(-2x)$$ Simplify: $$y' = (4 - x^2)^{1/2} - \frac{x^2}{(4 - x^2)^{1/2}} = \frac{4 - x^2 - x^2}{(4 - x^2)^{1/2}} = \frac{4 - 2x^2}{(4 - x^2)^{1/2}}$$ 4. **Step 3: Find critical points by setting $$y' = 0$$:** $$4 - 2x^2 = 0 \implies 2x^2 = 4 \implies x^2 = 2 \implies x = \pm \sqrt{2}$$ 5. **Step 4: Find corresponding y-values:** $$y(\pm \sqrt{2}) = \pm \sqrt{2} \cdot \sqrt{4 - 2} = \pm \sqrt{2} \cdot \sqrt{2} = \pm 2$$ 6. **Step 5: Find the second derivative to analyze concavity:** Using quotient and chain rules, the second derivative is $$y'' = \frac{d}{dx} \left( \frac{4 - 2x^2}{(4 - x^2)^{1/2}} \right)$$ After simplification, the concavity changes at points where $$y'' = 0$$. 7. **Step 6: Points of inflection occur where concavity changes, which is at $$x=0$$:** $$y(0) = 0$$ 8. **Summary:** - Local minimum at $$x = -\sqrt{2}, y = -2$$ - Local maximum at $$x = \sqrt{2}, y = 2$$ - Concave up on $$(-2, 0)$$ - Concave down on $$(0, 2)$$ - Point of inflection at $$(0, 0)$$ This matches the given information.