1. **State the problem:** We are given the function $f(x) = -2x^3$ and want to understand its behavior. 2. **Formula and rules:** The function is a cubic polynomial with a negative leading coefficient. Cubic functions have the general form $ax^3 + bx^2 + cx + d$. 3. **Intermediate work:** Here, $a = -2$, $b = 0$, $c = 0$, and $d = 0$. The function is odd and symmetric about the origin. 4. **Behavior:** Since $a$ is negative, the graph falls to the right and rises to the left. 5. **Derivative:** To find extrema, compute $f'(x) = -6x^2$. Since $f'(x) = 0$ only at $x=0$ and $f''(x) = -12x$, the point at $x=0$ is an inflection point, not a maximum or minimum. 6. **Intercepts:** The function passes through the origin $(0,0)$. **Final answer:** The function $f(x) = -2x^3$ has an inflection point at the origin and no local maxima or minima.