Answer
See graph and explanations.
Work Step by Step
Step 1. Given $y=x^3-12x^2$, we have $y'=3x^2-24x=3x(x-8)$ and $y''=6x-24=6(x-4)$
Step 2. The inflection points can be found at zeros of $y''$ where there is a sign (concavity) change. We have $..(-)..(4)..(+)..$ and we can identify an inflection point at $x=4$.
Step 3. The extrema can be found at zeros of $y'$ with sign changes in $y'$: $..(+)..(0)..(-)..(8)..(+)..$ and we can identify a local maximum at $(0,0)$ and a local minimum at $(8,-256)$.
Step 4. Graph the function and its derivatives as shown in the figure.
Step 5. The intersection of $y'$ with the x-axis indicates possible extrema positions.
Step 6. The intersection of $y''$ with the x-axis indicates possible inflection points.
Step 7. Positive $y'$ indicates increasing region and negative $y'$ indicates a decreasing region of the function.
Step 8. Positive $y''$ indicates concave up region and negative $y''$ indicates a concave down region of the function.
