The graphs of two functions are presented without axes.

The left hand function is increasing and concave down, begins decreasing and being concave down at a rounded corner, passes through a red marked point, is decreasing and concave up, and is increasing and concave up after a rounded corner.

The right hand function is decreasing and concave up, passes through a red marked point, and then is decreasing and concave down.

There is a label reading “Inflection Points” with arrows from it pointing at both red marked points.

$f^{″}(x)=0$ when $x=1$ and $x=-1$.

$f^{″}(x)$ does not exist when $x^3=0$ (i.e., when $x=0$).

Break up the domain of $f$ at each value found in Steps 2 and 3. Plug one number from each subinterval into $f^{″}(x)$ to determine the sign of $f^{″}(x)$ on that interval.

Notice that $f^{″}(x)$ changes sign at $x=-1$, $x=0$, and $x=1$, however, $x=0$ is not in the domain of $f$ and cannot correspond to an inflection point. Therefore, $(-1,-8)$ and $(1,8)$ are the only inflection points of $f$.

Find inflection points, if any, and determine if each one corresponds to a point of diminishing return, increasing return, or neither.

Notice that $f^{″}(x)=0$ only when $x=2$. Furthermore, $f^{\prime }(2)$ is positive, which means that $f(x)$ is increasing near $x=2$. And lastly, since $f^{″}(x)$ changes sign from negative to positive at $x=2$, $f(x)$ has a point of increasing returns at $x=2$.