Answer
See graph and explanations.
Work Step by Step
Step 1. Given $y=x^5-5x^4-240$, we have $y'=5x^4-20x^3=5x^3(x-4)$ and $y''=20x^3-60x^2=20x^2(x-3)$
Step 2. The inflection points can be found at zeros of $y''$ where there is a sign (concavity) change. We have $..(-)..(0)..(-)..(3)..(+)..$ and we can identify an inflection point at $x=3$ but not at $x=0$.
Step 3. The extrema can be found at zeros of $y'$ with sign changes in $y'$: $..(+)..(0)..(-)..(4)..(+)..$ and we can identify a local maximum at $(0,-240)$ and a local minimum at $(4,-496)$.
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 an increasing region and negative $y'$ indicates a decreasing region of the function.
Step 8. Positive $y''$ indicates the concave up region and negative $y''$ indicates the concave down region of the function.
