Answer
See graph and explanations.
Work Step by Step
Step 1. Given $y=\frac{4}{5}x^5+16x^2-25$, we have $y'=4x^4+32x=4x(x^3+8)$ and $y''=16x^3+32=16(x^3+2)$
Step 2. The inflection points can be found at zeros of $y''$ where there is a sign (concavity) change. We have $..(-)..(-\sqrt[3] 2)..(+)..$ and we can identify an inflection point at $x=-\sqrt[3] 2$.
Step 3. The extrema can be found at zeros of $y'$ with sign changes in $y'$: $..(+)..(-2)..(-)..(0)..(+)..$ and we can identify a local maximum at $(-2,13.4)$ and a local minimum at $(0,-25)$.
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 decreasing region of the function.
Step 8. Positive $y''$ indicates concave up region and negative $y''$ indicates concave down region of the function.
