1. **Stating the problem:** We are given the function $$f(x) = \sqrt[3]{x} e^x$$ and asked to identify its graph among the given options. 2. **Formula and rules:** The function is a product of two parts: the cube root function $$\sqrt[3]{x}$$ and the exponential function $$e^x$$. - The cube root function $$\sqrt[3]{x}$$ is defined for all real $$x$$ and is increasing, passing through the origin. - The exponential function $$e^x$$ is always positive and increases rapidly for positive $$x$$. 3. **Behavior analysis:** - For negative $$x$$, $$\sqrt[3]{x}$$ is negative but $$e^x$$ is positive but less than 1, so $$f(x)$$ is negative but close to zero. - At $$x=0$$, $$f(0) = \sqrt[3]{0} e^0 = 0 \cdot 1 = 0$$. - For positive $$x$$, both $$\sqrt[3]{x}$$ and $$e^x$$ are positive and increasing, so $$f(x)$$ increases rapidly. 4. **Graph features:** - The graph passes through the origin. - Slightly negative values for $$x<0$$, close to zero but negative. - Rapid increase for $$x>0$$. - No vertical asymptotes since $$\sqrt[3]{x}$$ is defined everywhere. 5. **Conclusion:** The correct graph is the one that passes through the origin, is slightly below the x-axis for negative $$x$$, and rises steeply for positive $$x$$ without vertical asymptotes. Final answer: The graph matching these features is the one described as "a smooth increasing curve passes near the origin, stays slightly below the x-axis for negative x, has a minimum around x ≈ -1, then rises steeply for x > 0".