1. The problem is to plot the function $y = x^3 + 3x$. 2. This is a cubic polynomial function where the highest degree term is $x^3$. 3. To understand the shape of the graph, we can analyze its critical points by finding the derivative: $$y' = \frac{d}{dx}(x^3 + 3x) = 3x^2 + 3$$ 4. Set the derivative equal to zero to find critical points: $$3x^2 + 3 = 0 \implies x^2 = -1$$ 5. Since $x^2 = -1$ has no real solutions, there are no real critical points, meaning the function has no local maxima or minima. 6. The function is strictly increasing or decreasing depending on the sign of the derivative. Since $3x^2 + 3 > 0$ for all real $x$, the function is strictly increasing everywhere. 7. The function passes through the origin $(0,0)$ and behaves like $x^3$ for large $|x|$. 8. The graph will have one inflection point where the concavity changes, found by the second derivative: $$y'' = \frac{d}{dx}(3x^2 + 3) = 6x$$ 9. Set $y''=0$ to find the inflection point: $$6x = 0 \implies x=0$$ 10. At $x=0$, the concavity changes, confirming the inflection point at the origin. Final answer: The function $y = x^3 + 3x$ is strictly increasing with an inflection point at $(0,0)$ and no local maxima or minima.