To get the list of possible inflection points for the function we’ll need the 2^nd derivative of the function so here that is.

\[\begin{align*}R'\left( z \right) & = {\left( {z + 4} \right)^{\,{\frac{2}{3}}}} + z\left( {{\frac{2}{3}}} \right){\left( {z + 4} \right)^{\, - \,\,{\frac{1}{3}}}} = \frac{{5z + 12}}{{3{{\left( {z + 4} \right)}^{{\frac{1}{3}}}}}}\\ R''\left( z \right) & = \frac{{5\left( {3{{\left( {z + 4} \right)}^{{\frac{1}{3}}}}} \right) - \left( {5z + 12} \right){{\left( {z + 4} \right)}^{ - {\frac{2}{3}}}}}}{{{{\left[ {3{{\left( {z + 4} \right)}^{{\frac{1}{3}}}}} \right]}^2}}} = \frac{{\left[ {15\left( {z + 4} \right) - \left( {5z + 12} \right)} \right]{{\left( {z + 4} \right)}^{ - {\frac{2}{3}}}}}}{{9{{\left( {z + 4} \right)}^{{\frac{2}{3}}}}}} = \underline {\frac{{10z + 48}}{{9{{\left( {z + 4} \right)}^{{\frac{4}{3}}}}}}} \end{align*}\]

Note that we simplified the derivatives at each step to help with the next step. You don’t technically need to do this but having the 2^nd derivative in its “simplest” form will definitely help with getting the answer to this part.

Now, recall that possible inflection points are where the 2^nd derivative either doesn’t exist or is zero. Because we simplified the 2^nd derivative as much as possible it is clear that the 2^nd derivative won’t exist at \(z = - 4\) (and the function exists at this point as well!). It should also be clear that the 2^nd derivative is zero at \(z = - \frac{48}{10} = - \frac{24}{5}\).

The possible inflection critical points of this function are then,

\[\underline {z = - \frac{24}{5} = - 4.8\,\,\,\,\,\,\,\,\& \,\,\,\,\,\,\,\,z = - 4} \]

There isn’t much to this part. All we really need here is a number line for the 2^nd derivative. Here that is,

From this we get the following concave up/concave down information for the function.

\[\require{bbox} \bbox[2pt,border:1px solid black]{{\,\,\,\,\,\,\,{\mbox{Concave Up : }}\left( { - \frac{24}{5}, - 4} \right)\,\,\,\,\,\& \,\,\,\,\,\left( { - 4,\infty } \right)\,\,\,\,\,\,\,\,\,\,\,\,\,{\mbox{Concave Down : }}\left( { - \infty , - \frac{24}{5}} \right)}}\]

For this part all we need to do is interpret the results from the previous step. Recall that inflection points are points where the concavity changes (as opposed to simply the points where the 2^nd derivative is zero or doesn’t exist). Therefore, the only inflection point for this function is,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{z = - \frac{24}{5}}}\]